Contact Joyce conjecture for oriented Legendrians

Let $(X,\mathcal{L},\alpha)$ be an oriented $-1$-shifted contact derived stack over $\mathbb{K}$, and let $\varphi\colon L\to X$ be a proper oriented Legendrian. Write

$$\operatorname{vdim}(\varphi)=\operatorname{vdim}(L)-\operatorname{vdim}(X)$$

for the relative virtual dimension of $\varphi$, and let $\mathcal{P}_X$ denote the associated perverse sheaf. Contact Joyce conjecture. There exists a morphism

$$\mu_L\colon \mathbb{F}_L[\operatorname{vdim}(\varphi)]\to \varphi^!\mathcal{P}_X$$

in $D_c^b(L_{\mathrm{lis\text{-}\acute{e}t}},\mathbb{F})$, whose local models in equivariant Darboux charts commute with the tame monodromy operator $T$. This conjecture is intended to provide the morphisms needed for $\ell$-adic integration and composition functors for Legendrian correspondences; the supplied text does not state whether it is proved or remains open.