Are respirators fitted with HEPA or P-100 filtered cartridges safe for DIY work, possibly in the presence of asbestos in the attic and roof? I'm in California, USA, but this Minnesota (USA) Department of Health page states:

Respirators must be equipped with HEPA filtered cartridges (color
coded purple) or an N-100, P-100 or R-100 NIOSH rating. These
cartridges are specific for filtering out asbestos fibers.

These filter 99.97% of airborne particles that are 0.3 microns or larger in diameter. However, this Occupational Safety and Health Administration (OSHA) page states, regarding asbestos exposure and sample fiber counting using Phase Contrast Microscopy (PCM):

A further disadvantage of PCM is that the smallest visible fibers are
about 0.2 um in diameter while the finest asbestos fibers may be as
small as 0.02 um in diameter. For some exposures, substantially more
fibers may be present than are actually counted.

A study, entitled Size and shape of airborne asbestos fibres in mines and mills, states regarding asbestos fiber sizes during industrial fiber processing:

The median true diameters of airborne crocidolite, amosite, and
chrysotile fibres, as determined by transmission electron microscopy,
were 0.07 μm (initial stage) to 0.09 μm (final stage), 0.20 μm
(initial) to 0.26 μm (final), and 0.05 μm (initial) to 0.06 μm (final)
respectively…

Many asbestos fibers are much smaller than 0.3 μm in diameter. Does this mean there's no safe respirator filter, whether HEPA, P100, or better, to protect against asbestos? Do better air-purifying respirator filters exist? Or, is using an atmosphere-supplying respirator the only way to be safe?

EDIT: piojo's answer shows the HEPA spec requires 99.97% filter efficiency for 0.3 μm diameter particles, not for 0.3 μm diameter "or larger" particles. His source describes filter efficiency as the aggregate efficiency of several mechanisms: sieving, inertial impaction, interception, and [Brownian] diffusion. The most penetrating particle size (MPPS) is where aggregate efficiency is weakest. It states 0.3 μm is the benchmark because it is near that size. So, 99.97% efficiency is near the MPPS efficiency. Its graph assumes ≥99.97% efficiency for all particles down to 0.01 μm, but I've found no source that corroborates this efficiency for sub-0.07 μm, which is well within the range of airborne asbestos diameters. Efficiency should approach 100% with increasing diameters beyond the MPPS, due to increasing effectiveness of interception, impaction, and sieving. While efficiency increases with diameters immediately smaller than the MPPS due to diffusion, efficiency can't keep increasing when below a certain diameter, because the filter must pass air without making it too hard to breathe—the efficiency will drop below MPPS efficiency at some small diameter. MPPS efficiency should be confined between infinity and this small diameter. I've found no source that identifies this limit.

Wikipedia says the MPPS is 0.21 μm, citing the research paper Particle Size for Greatest Penetration of HEPA Filters—and Their True Efficiency. It's dated 1982, so I don't know if the HEPA spec and filter composition have since changed. The paper doesn't identify any percentage efficiency at 0.3 μm required by 1982's HEPA spec. The paper states:

The penetration at 0.21 μm is calculated to be seven times greater
than at the 0.3 μm used for testing HEPA filters.

Much of it discusses efficiency of a single fiber of the filter, so I don't know how whole-filter efficiency scales. I don't know if 1982's HEPA 0.3 μm efficiency spec is 99.97% nor if whole-filter efficiency scales the same as single-fiber efficiency, i.e. I don't know if modern whole-filter MPPS efficiency is:

99.97% - [7 * (100% - 99.97%)] = 99.76% ??

The paper collected data from other research, adopted some of their assumptions and made its own, and extrapolated to establish single-fiber efficiency curves. In many of these curves, the experimental data points do not fit well with the curves it's trying to theorize. For example, Figure 4's theoretical curve is significantly translated upwards (towards higher efficiency) from the empirical data points. It glosses over the reasoning with: "The conversion was made from protection factor, P.F., to single-fiber efficiency…" but never defines P.F. or mentions it again. Even if the curve weren't translated, its shape won't fit well with an actual best-fit curve through those points. It only has 6 empirical data points, the smallest particle diameter of which is 0.07 μm, which is around the median diameter of airborne crocidolite particles and larger than the median diameter of airborne chrysotile particles. Figure 3 has no experimental data for sub-0.1 μm diameters. Yet, both Figures jump to conclusions by extrapolating ill-fitting curves to 0.02 μm. These are the only efficiency vs. diameter data this paper has on HEPA filters. Most of the data this paper uses to determine HEPA filter's MPPS (called SOMP therein) comes from non-HEPA filters!

The wiki lists electrostatic attraction as a filtration mechanism, but lacks a citation. The paper doesn't mention this mechanism, but it states that filters increase in efficiency with use due to loading (i.e. getting filled with particles), up to a certain extent before the trapped particles slip through (page 13).

Another research paper, entitled Fiberglass Vs. Synthetic Air Filtration Media, states that loading reduces electrostatic filtration effectiveness due to decreasing the filter's exposed surface area and that aerosols neutralize electrostatic charges. I don't know how well electrostatically charged filters hold asbestos particles compared to other particles of the same diameter. How would aerosol fit-tests (e.g. nebulizing saccharin or Bitrex) test for electrostatic filtration? Would the tests decrease the filter's efficacy?

EDIT 2: The research paper Total Inward Leakage of Nanoparticles Through Filtering Facepiece Respirators shows a model-unspecified P100 filter on a full-face respirator that's perfectly sealed to the face (using silicone sealant) to have >99.97% efficiency for 8 different diameters of charge-neutralized particles in the 0.08–0.4 μm range (Figure 3). The graph isn't a best-fit curve. It connects points with straight lines. Because the MPPS isn't plotted, the graph doesn't show its dip in efficiency. The author clarifies by stating: "Particle penetration was several times higher for the particles in the MPPS than for 8 and 400 nm size ranges. P100 FFR does not show the MPPS at 40–50 nm range in the figure because of the low penetration value (<0.03%)." Because the author implies his test filter has >99.97% efficiency at the 0.04–0.05 μm MPPS and this MPPS is within his 0.08–0.4 μm test range, he implies the filter has >99.97% efficiency over the 0.08–0.4 μm range. The author doesn't specify the filter's model number. Is this performance representative of other filters? Does every P100 filter's per-HEPA-spec ≥99.97% efficiency at 0.3 μm guarantee ≥99.97% efficiency for the entire 0.08–0.4 μm range? Note that this paper appears to use a filter that greatly surpasses the HEPA spec, because the spec expects the MPPS to be near 0.3 μm, whereas this paper's is much smaller.