Microwave sponges or not?

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.

Both of the original linked articles/blogs are wrong in some respect and represent an over-dramatization and mis-interpretation of the results in both studies, but particularly in the case of the Scientific Reports study.

The Good Housekeeping one was performed in a registered analytical laboratory, and while the direct results are not published, there is a good chance that they are correct; using a method of cleaning reduces the viability of microbes in your sponge to some degree. The data they present suggests that bleach (a very good chemical sanitizer) as expected, kills >99.9% of the bacteria, as does microwaving (heat) and the dishwasher (heat again; note the heated dry cycle mentioned). Note that the figures quoted such as 99.9% are all about the certainty of the limits of the tests; it may be that the test actually killed 99.999999% of bacteria, but the test won't be precise enough to measure with that degree of precision.

Now just how much is 99.9% - well, that's 999 bacteria out of 1000 or one viable out of every 1000. 1/1000 sounds great, but bacteria are super abundant on most things, so in reality its more like 1000 bacteria left out of 1 million. There's a good article here (paywalled?), at International Journal of Food Microbiology, which studied abundance of bacteria in sponges which inoculated sponges and left them for several days to see how many bacteria grew. They found that after 3 days, up to 10⁹ (that's 1 billion) bacterial colonies grew.

The Apartmenttherapy page badly misinterpreted the article it is based on. The published Scientific Reports article, while it is peer-reviewed and published (both good things from a scientist's perspective), makes no assertions about the cleaning methods at all or the viability of the microbes after cleaning. It used incredibly sensitive methods of molecular detection to determine the abundance (meaning number of species not number of bacteria) in each sponge. It also looked at the abundance of bacteria by a molecular method, which can be used to infer numbers. However, this method can not distinguish between viable bacteria and non-viable bacteria.

There are a number of papers that have directly looked at decontamination of sponges and almost all of them found that cleaning works to a greater or lesser extent depending on the method of cleaning. Here are a couple of commonly cited ones:

https://www.scirp.org/journal/paperinformation.aspx?paperid=20492 (open access) https://www.sciencedirect.com/science/article/pii/S0956713508001667?via%3Dihub (paywall?)

Yes, cleaning your sponge gets rid of some of the bacteria by making them non-viable, but there will be some remaining - no matter the cleaning method (unless you happen to have access to something like an autoclave). The remaining bacteria grow rapidly - under the ideal conditions, in a lab, something like Escherichia coli, a common gut bacterium can double in population number in 20 min at logarithmic growth, so if you started with 1 bacterium at time 0, after 24 hours you could have 2⁷² or 4.7 x10²¹ bacteria (4.7 thousand trillion). This number would never happen in reality (as seen in the IJFM article linked above), as there are all sorts of constraints on this sort of growth in real life - space, food sources, temperature etc, but the point stands

Cleaning works to a point, but change your sponges frequently anyway.