Background
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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 $Psh(A)$ satisfying the following axioms:

1.) The class W satisfies 2-for-3

2.) The class $W$ contains $\mathrm{rlp}(\mathrm{Mono}(A))$, where $\mathrm{Mono}(A)$ denotes the class of all monomorphisms in $Psh(A)$.

3.) The class $W\cap Mono(A)$ is closed under transfinite composition and pushouts.

Given a class $C$ of morphisms in $Psh(A)$, 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 $Psh(A\times \Delta)$ as $Psh(A)^{\Delta^{op}}$) together with the class of all maps $T\otimes \Delta^1\to T$ where $T$ is an object of $Psh(A\times \Delta)$ (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 $pr_1^\ast:Psh(A)\to Psh(A\times \Delta)$ obtained from the projection onto the first factor induces an equivalence of categories $W^{-1}Psh(A) \to W_\Delta^{-1}Psh(A\times \Delta)$. 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 $(Psh(A), Mono(A), W)$ and $(Psh(A\times \Delta),Mono(A\times \Delta), W_\Delta$.  

Recall that the injective (with respect to $\Delta$) model structure on $Psh(A\times\Delta)\cong Psh(\Delta)^{A^{op}}=sSet^{A^{op}}$ is the model structure for which the cofibrations are exactly the monomorphisms and the class of weak equivalences $W_{inj}$ comprises the levelwise weak homotopy equivalences.  

We will say that an $A$-localizer $W$ is a _Bousfield localization_ of another $A$-localizer $W^\prime$ if $W^\prime \subseteq W$.  

Question:
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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 $W_{inj}$?

Remarks
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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 $sSet^{A^{op}}$ coincides with the injective model structure.