I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $\mathrm{Top}$ of $\Delta$-generated topological spaces where $\mathcal{M}$ has for objects the set of strictly positive real numbers and where the space $\mathcal{M}(L,L')$ is the space of nondecreasing surjective maps from $[0,L]$ to $[0,L']$.

An important class of examples of $\mathcal{M}$-spaces is what I call the quasicofibrant $\mathcal{M}$-spaces. Roughly, an $\mathcal{M}$-space $X$ is quasicofibrant when it really comes from a space of continuous paths in a topological space and if the maps of the diagram $X$ are really reparametrization maps.

Consider a topological space $U$, a set of continuous paths $P$ from $[0,1]$ to $U$ going from $\alpha$ to $\beta$ fixed such that $P$ is closed by reparametrizations by the maps of $\mathcal{M}(1,1)$; then one can define from $P$ an $\mathcal{M}$-space $X(P)$ by $X(P)(L)$ being the space of composites $\gamma\phi$ where $\gamma\in P$ and $\phi\in \mathcal(L,1)$. Note that $X(P)(1)=P$. Such an $\mathcal{M}$-space $X(P)$ is by definition quasicofibrant.

The terminology quasicofibrant comes from the following fact. Let us equip $[\mathcal{M}^{op},\mathrm{Top}]_0$ with the projective q-model structure : the fibrations are the objectwise q-fibrations and the weak equivalences are the objectwise weak homotopy equivalences.

Proposition: Every projective q-cofibrant $\mathcal{M}$-space is quasicofibrant. The converse is false.

There is a biclosed semimonoidal structure on $[\mathcal{M}^{op},\mathrm{Top}]$ defined as follows: $$D\otimes E = \int^{(L_1,L_2)} D(L_1)\times E(L_2) \times \mathcal{M}(-,L_1+L_2).$$

Proposition: Let $f:A\to B$ and $g:C\to D$ be two objectwise weak homotopy equivalences. Assume that $A,B,C,D$ are projective q-cofibrant. Then $f\otimes g$ is an objectwise weak homotopy equivalence.

The proof goes as follows. We write $f \otimes g = (f\otimes 1_D) (1_A \otimes g)$. It then suffices to prove e.g. that $1_A \otimes g$ is a weak equivalence. To conclude, it suffices to observe that the functor $A\otimes -:[\mathcal{M}^{op},\mathrm{Top}]_0\to [\mathcal{M}^{op},\mathrm{Top}]_0$ is a left Quillen adjoint. This relies on Theorem 9.4 of my paper https://doi.org/10.32408/compositionality-3-3.

The question is

Is the tensor product of two objectwise weak homotopy equivalences of $\mathcal{M}$-spaces an objectwise weak homotopy equivalence of $\mathcal{M}$-spaces ?

I am not convinced that it is true in full generality. However, I suspect that it is true by replacing the hypothesis $A,B,C,D$ projective q-cofibrant by the hypothesis $A,B,C,D$ quasicofibrant :

Is the tensor product of two objectwise weak homotopy equivalences between quasicofibrant $\mathcal{M}$-spaces an objectwise weak homotopy equivalence of $\mathcal{M}$-spaces ?