Suppose we are given categories $\mathsf{C},\mathsf{D},\mathsf{E},$ equipped with collections of weak equivalences $\mathcal{W}_{\mathsf{C}},\mathcal{W}_{\mathsf{D}},$ and $\mathcal{W}_{\mathsf{E}},$ respectively. Suppose that we are also given functors $F : \mathsf{C}\to\mathsf{E}$ and $G : \mathsf{D}\to\mathsf{E}$ which send weak equivalences to weak equivalences. Denote by $\mathsf{C}\overset{\rightarrow}{\times}_{\mathsf{E}}\mathsf{D}$ the "directed fiber product" (i.e., the comma category of triples $(c\in\mathsf{C},d\in\mathsf{D},\alpha : F(c)\to G(d)\in\mathsf{E})$). We may equip $\mathsf{C}\overset{\rightarrow}{\times}_{\mathsf{E}}\mathsf{D}$ with a collection of weak equivalences $\mathcal{W}$ given by morphisms $$(f,g) : (c,d,\alpha)\to(c',d',\alpha')$$ such that $f\in\mathcal{W}_{\mathsf{C}}$ and $g\in\mathcal{W}_{\mathsf{D}}.$ Let $L(\mathsf{C},\mathcal{W}_{\mathsf{C}})$ denote the $\infty$-categorical localization/underlying $\infty$-category of $\mathsf{C}$ with respect to $\mathcal{W}_{\mathsf{C}}$ (and similarly for the other categories).

**My Question:** Is there an equivalence of $\infty$-categories
$$
L(\mathsf{C},\mathcal{W}_{\mathsf{C}})\overset{\rightarrow}{\times}_{L(\mathsf{E},\mathcal{W}_{\mathsf{E}})}L(\mathsf{D},\mathcal{W}_{\mathsf{D}})\simeq L(\mathsf{C}\overset{\rightarrow}{\times}_{\mathsf{E}}\mathsf{D},\mathcal{W}),
$$
and if not, are there any reasonable conditions under which this would hold?

I'm happy to assume that all of the categories in question are proper, simplicial, cellular model categories and that the weak equivalences are the weak equivalences in these model structures, and even that $\mathsf{C} = \mathsf{E},$ $\mathcal{W}_{\mathsf{C}} = \mathcal{W}_{\mathsf{E}},$ and that $F = \operatorname{id}.$

I'm aware that this holds for $1$-categorical localizations of products of model categories, as in this question, and I presume the statement for products also holds when we consider $\infty$-categorical localizations of the model categories as well via a similar argument (although I haven't checked this carefully -- please correct me if I'm wrong on this point).

My motivation for this question is that I have encountered a model category that I'm interested in, described as a "directed fiber product" with a model structure as in the question. I want work with the underlying $\infty$-category of this model category, but it would be much easier to do so if I know that I can work with $\infty$-categories from the start to build the category I'm interested in, and avoid the model-category details as much as possible.