Background
Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following axioms:
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The class $W$ satisfies 2-for-3
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The class $W$ contains $\mathrm{rlp}(\mathrm{Mono}(A))$, where $\mathrm{Mono}(A)$ denotes the class of all monomorphisms in $Psh(A)$.
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The class $W\cap \mathrm{Mono}(A)$ is closed under transfinite composition and pushouts.
Given a class $C$ of morphisms in $\mathrm{Psh}(A)$, define $W(C)$, the localizer generated by $C$, to be the intersection of all $A$-localizers containing $C$. We say that a localizer $W$ is accessible if there exists a small set $S$ such that $W=W(S)$.
Let $(A,W)$ be a pair comprising a small category $A$ and an $A$-localizer $W$. We define the simplicial completion of $(A,W)$ to be the pair $(A\times \Delta, W_\Delta)$, where we define the $A\times \Delta$-localizer $W_\Delta$ as follows: The class $W_\Delta$ is the localizer generated by $W^{\Delta^{op}}$, the class of levelwise $W$-equivalences (viewing $\mathrm{Psh}(A\times \Delta)$ as $\mathrm{Psh}(A)^{\Delta^{op}}$, so the levels are indexed by the objects of $\Delta$) together with the class of all maps $T\otimes \Delta^1\to T$ where $T$ is an object of $\mathrm{Psh}(A\times \Delta)$ (given an object $T$ and a simplicial set $X$, we define $(T\otimes X)(a,s)=T(a,s)\times X(s)$ for a pair of objects $a$ in $A$ and $s$ in $\Delta$).
It is a theorem of Cisinski that the functor $pr_1^\ast:\mathrm{Psh}(A)\to \mathrm{Psh}(A\times \Delta)$ obtained from the projection onto the first factor induces an equivalence of categories $W^{-1}Psh(A) \to W_\Delta^{-1}\mathrm{Psh}(A\times \Delta)$. Further, if the $A$-localizer $W$ is accessible, the functor $pr_1^\ast$ is the left Quillen functor of a Quillen equivalence between the model categories $(\mathrm{Psh}(A), \mathrm{Mono}(A), W)$ and $(\mathrm{Psh}(A\times \Delta),\mathrm{Mono}(A\times \Delta), W_\Delta)$.
Recall that the injective (with respect to $\Delta$) model structure on $\mathrm{Psh}(A\times\Delta)\cong \mathrm{Psh}(\Delta)^{A^{op}}=\mathrm{sSet}^{A^{op}}$ is the model structure for which the cofibrations are exactly the monomorphisms and the class of weak equivalences $W_{\mathrm{inj}}$ comprises the levelwise (indexed by the objects of $A$) weak homotopy equivalences.
We will say that an $A$-localizer $W$ is a Bousfield localization of another $A$-localizer $W^\prime$ if $W^\prime \subseteq W$ (because if $W$ and $W^\prime$ are accessible, the model structure associated with $W$ is a left Bousfield localization of the model structure associated with $W'$.
Question:
Given a small category $A$ and an $A$-localizer $W$, is the $A\times \Delta$-localizer $W_\Delta$ a Bousfield localization of the class of levelwise weak homotopy equivalences $W_{\mathrm{inj}}$?
Remarks
I can prove that this is true in the special case that $A$ has the following property:
The category $A$ is a Reedy category such that the Reedy model structure on $sSet^{A^{\mathrm{op}}}$ coincides with the injective model structure. Somewhat surprisingly, this special case happens to include some of the most interesting examples. Cisinski has given an axiomatic characterization of a wide swath of these categories using his formalism of "catégories squelletiques" (Astérisque 308 ch. 8). Also see this MO Question.
Best Answer
Let $A$ be a small category, and $W$ an $A$-localizer. Then we say that $W$ is regular if any presheaf $X$ over $A$ is canonically the homotopy colimit of the representable presheaves above $X$; see Definition 3.4.13 (all references are in Astérisque 308). Except stated otherwise, all the assertions below about regular $A$-localizers are proved in section 3.4.
Given any $A$-localizer $W$, there is a minimal regular $A$-localizer containing $W$. This process of regularization preserves all the interesting properties of $W$ you might expect (namely, being accessible, proper, cartesian, being closed under small filtered colimits, respectively). What is nice is that there is the following
Theorem 3.4.36. An $A$-localizer $W$ is regular if and only if its simplicial completion $W_\Delta$ contains the class $W_{inj}$ of termwise simplicial weak equivalences.
This has useful consequences. For instance, any regular $A$-localizer is closed under small filtered colimits. There is also a kind of internal relative version of Quillen's theorem A. See Corollaries 3.4.41 and 3.4.47.
There is a whole class of squelettique categories $A$ for which any $A$-localizer is regular; see Proposition 8.2.8 (the proof I give relies on the general results of regular localizers, but there is a more elementary proof). However, for this property to hold, you need all the automorphisms of objects of $A$ to be trivial. Indeed, you may find in Proposition 3.4.57 that equivariant homotopy theory (à la Peter May) gives a nice example of a squelettique category $A$ with a nice explicit complete homotopical structure whose associated $A$-localizer $W$ is proper, cartesian, closed under filtered colimits, but not regular (in particular, its simplicial completion does not contain the class of termwise simplicial weak equivalences). This $W$ also has the property that any map between representable presheaves is a weak equivalence, and that, for any representable $a$, the associated $A/a$-localizer is regular (in this example, the objects of $A/a$ have not any non trivial automorphisms). In particular, this shows that the characterization of the $A$-localizer of $\infty$-equivalences for a (local) test category given by Proposition 6.4.26 is the optimal one (at least in this language).
N.B. This kind of question has also applications in comparing the various notions of $\infty$-categories. For instance, the Quillen equivalence between Joyal's model category for quasi-categories and Rezk's model category for complete Segal spaces can be explained rather immediately from the correspondance $W\mapsto W_\Delta$ and from the fact that any $\Delta$-localizer is regular (and using Theorem 3.4.36 above). The same thing will happen for the analogs for $(\infty,n)$-categories by the same argument.