I can't decide between the conflicting research! Who's correct?
Stop Microwaving Your Sponges, Immediately.
“When people at home try to clean their sponges, they make it worse,” Egert said — similar to how people can encourage antibiotic resistant bacteria if they don’t follow the doctor’s orders.", from a Seattle Times article
Our research suggests that long term cleaning might select for potentially pathogenic and/or smelly bacteria. We think this is because some bacteria can adapt to the cleaning process, survive the microwave or dishwasher, and can easily grow to higher numbers again. from a blog
The microwave was one of the next most effective, zapping 99.9% of germs.
"Basically, what we find is that we could knock out most bacteria in two minutes," says researcher Gabriel Bitton, professor of environmental engineering at the University of Florida, in a news release. "People often put their sponges and scrubbers in the dishwasher, but if they really want to decontaminate them and not just clean them, they should use the microwave." from WebMD
Best Answer
The question in the title is unanswerable. Your own search shows that there is no simple prescription of what you should do.
The question "who is correct" is: both of them.
The side you are interpreting as "you should microwave sponges" tells you that microwaving kills most of the bacteria. The side you are interpreting as "you shouldn't microwave sponges" tells you that, after you have microwaved the sponges, the next generation of bacteria that grows in them is harder to kill, and potentially more dangerous to your health. Both statements are completely compatible with each other.
What is wrong is to interpret these tiny pieces of data as a general prescription for optimal behavior. They are simply not sufficient to draw any conclusions from that kind. And the way it looks, nobody has done further research (which would be immensely complicated) to determine whether it is better to microwave or not microwave, and if you do microwave, then in what pattern. It would probably take several professors' full careers to settle such a question (because you would need longitudinal studies too), and I doubt that there is enough public interest to finance that, seeing that nobody has identified a major public health problem stemming from washing your dishes with a sponge.
So in the end, you will have to base your personal decision to microwave or not, and base it on some criteria other than which one presents a higher risk to your health, since that information is presently unknowable.