In “Algebras and Modules in Monoidal Model Categories” Schwede and Shipley introduced the monoid axiom. If a cofibrantly generated monoidal model category $M$ satisfies this axiom and some smallness hypotheses, then the category $Mon(M)$ of monoids inherits a model structure with fibrations and weak equivalences taken from $M$. In that paper and related papers by the same authors, the monoid axiom is shown to hold on sSet, simplicial functors, simplicial abelian groups, $\Gamma$-spaces, symmetric spectra, $S$-modules, orthogonal spectra, Ch(R), and StMod(R).

The monoid axiom was explored further in Hovey's preprint “Monoidal Model Categories.” Here he showed it to hold on compactly generated spaces and he remarks on page 5 that he doesn't know if it holds on $K$-spaces. The proof he gives for compactly generated spaces fails for $K$-spaces because compact spaces are not ﬁnite relative to closed inclusions, but only relative to closed $T_1$ inclusions. This was about 10 years ago, so I have to ask:

Does the monoid axiom hold on $K$-spaces?

I've read in several papers that there is no known model structure on the category of topological monoids. For instance, here is one written by Vogt in 2012 which makes this claim. I don't know if that's because the monoid axiom is known to fail, or if it just isn't known, or if the obstruction to building a model structure has more to do with smallness than with the monoid axiom. This is what I'm trying to get at.

Is there some known obstruction to $Mon(K$-spaces$)$ being a model category? If not, could experts weigh in on why this is such a hard problem, or on whether or not they think it's true?

My interest in this is for writing the background section of my thesis. I do a lot in my thesis with monoids and commutative monoids, and deal a lot with the monoid axiom. Most of what I do probably won't have an application to categories of spaces, and that was not the motivating application. Still, it would be nice to figure out what my results say in that context (if anything) and that's also what motivated my recent questions on $W$-spaces (see here and here). Incidentally, it was also for this background section that I asked my other question on $K$-spaces and learned that they are not named for Kelley.

Incidentally, I know that some will say this is the wrong question. Instead of strict monoid structures we should care about $A_\infty$ structures. I'm aware of that argument and of the theory in the $A_\infty$ setting, but I have found there are still interesting things to say about strict monoids and strict commutative monoids. So let's restrict attention to that case and only bring in $A_\infty$ as it helps solve the question about strict monoids (e.g. by rectification). Note that this MO question strikes at the difference between $A_\infty$ and strict monoids for spaces.