If a swashbuckler with panache falls to 0 hp, do they lose panache? I can't find anything in the swashbuckler description that directly addresses it, but thematically it sure makes sense — hard to keep your swagger when you're gurgling to death on the ground.
Does a PF2 Swashbuckler lose panache if they go down
pathfinder-2eswashbuckler
Related Solutions
Returning does not mean Reload 0
Most weapons with the Thrown trait like the Dart, have a reload of "—". This means that the weapon is not reloaded, but rather drawn with an interact action, as mentioned in the reloading rules, then thrown.
The returning rune eliminates the need of drawing a new weapon to replace the one you just throw, so you can make as many strikes with your thrown weapon as your actions allow. This does not change its reload value (it it did, it would say so).
Double Shot feat designates ranged weapons with a explicit reload of 0, like longbows, shortbows and shurikens (the shuriken is a special case, the only thrown weapon with a listed reload value of 0). A reload of 0 means that the action of drawing ammunition and firing the weapon are part of the same action. The feat Double Shot can be read as drawing two projectiles (arrows or shurikens) at once, then making two consecutive strikes without stoping to reload.
So no, you cannot use Double Shot with thrown weapons other than shurikens. On the other hand, if you use shurikens, a literal reading of the rules says that you can indeed use the Double Shot feat (and the Triple shot feat) with a single returning shuriken, as it would fly back to you after each strike and in time for the next (how it would manage to do that in time when you are doing several strikes in "blindingly fast succession" is left to interpretation...).
The Acrobatics skill has untrained action: Balance. CRB p 240 describes how it works:
Requirements: You are in a square that contains a narrow surface, uneven ground, or another similar feature.
You move across a narrow surface or uneven ground, attempting an Acrobatics check against its Balance DC. You are flat-footed while on a narrow surface or uneven ground.
Depending on skill check result, the character moves, stays still or falls down. Since the floor is "just" transparent, maybe falling would mean that the character falls prone. On stairs and walkways there's real possibility to fail down, so the reaction Grab an Edge (CRB 472) might be able to save the day.
Maybe a Perception check would be reasonable way for the characters to counteract the mirrors. A character who succeeds in perception notes, say, dust on the floor and shadows or malformations in the reflection and is able to reason the right way.
There doesn't seem to be a rule about vertigo. Since it's a mental effect, that could call for a Will save. The save might provide +2/+1/-1/-2 to next perception check. Rationale is that if one don't suffer from vertigo effect, one can look around very carefully for telltales on the reflection. On the other hand, a character who's about to throw up lunch is in no condition to be particularly observant.
Since the puzzle relies on a bit of homebrew rules, you should simulate it and see how the characters would fare. Perception is a good skill for any adventurer, and will saves are important against several nasty effects, so neither should be overlooked on character development. Should the party be weak on these, the puzzle is likely to be more frustrating than entertaining a brain-teaser.
For higher-level characters who are capable to cast True See and Fly, the puzzle is likely to be trivial. For a low-level party, maybe it's a good idea to cut a bit of slack. The Glitterdust spell isn't RAW applicable, as it targets creatures. However, it would be a creative solution for parts of the puzzle. There aren't probably many casters who prepare Glitterdust more than once. Other spells affecting the environment, such as Web or Grease might be used to note the path too. Light up the grease, wait, and follow the sooth trails. Also, if the characters bring up stuff to cover the floor, be it dust, sand or whatever, consider rewarding the creative approach.
Best Answer
No
The panache rules are silent on this specific concern, yes, but there is no indication that any condition, dying included, causes them to lose Panache. Instead:
Per the rules, the only ways to lose Panache are spending it on a Finisher or having an encounter end. Dying meets neither of these criteria - neither does being unable to act through a variety of conditions (stunned/slowed 3, confused, etc.), nor does even an "uncool" action like Fleeing cause you to lose Panache.
You may not imagine dying as having a lot of swagger (which is a bit insulting to my overly bold heroic sacrifices to save my squishy comrades), but one of the narrative keys to Panache is that it is gained by a moment of swagger/style/cool, not by the continuous state of doing repeatedly cool things. Indeed, you don't ever need to make checks to maintain your Panache - you do something once and then leverage that moment later on. Even the description lines up with this:
Panache is capitalizing on a past moment, not on a continuous streak of moments. So the Swashbuckler, via RAW and RAI, should keep their Panache.
Also, rewarding cowardly self-defensive behavior instead of unnecessarily risky and reckless behavior would make this Swashbuckler player very sad, and mechanically Swashbucklers are already a little undertuned, so removing Panache from them for any reason is a very unnecessary nerf that prevents me from rising from dying, bleeding out and breathing raggedly, only to grab my sword and stab my foe with a sudden burst of confident vigor accompanied by a witty catchphrase ala "Your turn to die, fiend!". (While it's not exactly the same, if you've seen the Princess Bride, think of Inigo Montoya's sudden burst of vigor against the Six-Fingered Man, or Wesley's sudden burst of energy that intimidates Prince Humperdink into surrender).