This is a follow-up question to another question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and prove monoidal Quillen equivalences between them. One of the models is $W$-spaces, i.e. diagrams of shape $W$ in $Top$ where $W$ is the category of based spaces which are homeomorphic to finite CW complexes, and $Top$ means compactly generated spaces.

Does the category of commutative monoids in $W$-spaces inherit a model structure?

By commutative here I mean strictly commutative, not $E_\infty$. My interest is for the background section of a paper I'm working on now about when commutative monoids form a model category. If it's not known one way or the other I can try my theorem on the category of $W$-spaces, but I suspect that category won't satisfy my hypotheses. If it's known to be true, maybe I can recover it as a special case of my theorem. If it's known to be false, that would be good to know too.

Note that this question comes up on page 5 of MMSS and the authors say they don't know. On page 47 they point out the obstacle which prevented them from proving CommMon($W$-spaces) inherits a model structure, and it's the same obstacle Peter May pointed out in his answer to my previous question. They couldn't prove that every relative $Sym(K^+)$-cell complex was a stable equivalence (where $K^+$ are the positive stable generating trivial cofibrations) because they couldn't prove $W$-spaces have the property that for every cell $S$-module $M$, $(E\Sigma_j)_+ \wedge_{\Sigma_j} M^j \to M^j/\Sigma_j$ is a weak equivalence. If someone had proven this in the past decade I'm sure Peter May would know. What I'm wondering is if someone found a different way to prove CommMon($W$-spaces) forms a model category or if someone found evidence that it can't, similar to the evidence in this answer that CDGA can't be a model category, or on page 13 of this paper of Schwede's that CommMon($\Gamma$-spaces) can't be a model category. Even if no hard evidence exists, I'd like to hear some soft evidence, i.e. why one would or would not believe this to be true.

The reason one might hope the answer is yes is that Theorems 0.1 and 0.4 prove that $W$-spaces are Quillen equivalent to Orthogonal Spectra and Symmetric Spectra, and that this Quillen equivalence carries over when we pass to the model categories of monoids (the fact that such model structures exist follows from the fact that all these categories satisfy the pushout product and monoid axioms). Both Symmetric Spectra and Orthogonal Spectra have a notion of positive model structures, which makes the category of commutative monoids inherit a model structure. It's natural to wonder if $W$-spaces has something like that too.

MMSS also raise another natural question (on page 6):

Is $Ho(E_\infty$ $W$-spaces$) \cong Ho($Commutative $W$-spaces$)$?

It seems like with all the work that has been done on operads in the last decade this should be known. Can someone confirm that and give a reference?

This is a follow-up question to another question I asked last month. In MMSS's "Model Categories of Diagram Spectra," the authors consider many different models for spectra and prove monoidal Quillen equivalences between them. One of the models is $W$-spaces, i.e. diagrams of shape $W$ in $Top$ where $W$ is the category of based spaces which are homeomorphic to finite CW complexes, and $Top$ means compactly generated spaces.

Does the category of commutative monoids in $W$-spaces inherit a model structure?

By commutative here I mean strictly commutative, not $E_\infty$. My interest is for the background section of a paper I'm working on now about when commutative monoids form a model category. If it's not known one way or the other I can try my theorem on the category of $W$-spaces, but I suspect that category won't satisfy my hypotheses. If it's known to be true, maybe I can recover it as a special case of my theorem. If it's known to be false, that would be good to know too.

Note that this question comes up on page 5 of MMSS and the authors say they don't know. On page 47 they point out the obstacle which prevented them from proving CommMon($W$-spaces) inherits a model structure, and it's the same obstacle Peter May pointed out in his answer to my previous question. They couldn't prove that every relative $Sym(K^+)$-cell complex was a stable equivalence (where $K^+$ are the positive stable generating trivial cofibrations) because they couldn't prove $W$-spaces have the property that for every cell $S$-module $M$, $(E\Sigma_j)_+ \wedge_{\Sigma_j} M^j \to M^j/\Sigma_j$ is a weak equivalence. If someone had proven this in the past decade I'm sure Peter May would know. What I'm wondering is if someone found a different way to prove CommMon($W$-spaces) forms a model category or if someone found evidence that it can't, similar to the evidence in this answer that CDGA can't be a model category, or on page 13 of this paper of Schwede's that CommMon($\Gamma$-spaces) can't be a model category. Even if no hard evidence exists, I'd like to hear some soft evidence, i.e. why one would or would not believe this to be true.

The reason one might hope the answer is yes is that Theorems 0.1 and 0.4 prove that $W$-spaces are Quillen equivalent to Orthogonal Spectra and Symmetric Spectra, and that this Quillen equivalence carries over when we pass to the model categories of monoids (the fact that such model structures exist follows from the fact that all these categories satisfy the pushout product and monoid axioms). Both Symmetric Spectra and Orthogonal Spectra have a notion of positive model structures, which makes the category of commutative monoids inherit a model structure. It's natural to wonder if $W$-spaces has something like that too.

MMSS also raise another natural question (on page 6):

Is $Ho(E_\infty$ $W$-spaces$) \cong Ho($Commutative $W$-spaces$)$?

It seems like with all the work that has been done on operads in the last decade this should be known. Can someone confirm that and give a reference?