The nearby-cycles identification for visible stable polar representations

Let $(G,V)$ be a visible stable polar representation for a connected reductive group $G$. Let $\pi\colon V\to V/\!/G$ be the quotient map, let $P$ be its nearby cycles sheaf, let $\mathbb{L}$ be the rank-$|W|$ local system on $V_{\operatorname{reg}}$ whose minimal extension is the Fourier transform $\mathbb{F}^{*}_{V}(P)$, and let $j\colon V_{\operatorname{reg}}\hookrightarrow V$ be the inclusion. Assume that $A_{\kappa}=A_{\kappa}(W)$ is simple when $\varsigma=0$. Nearby-cycles identification. There is an isomorphism

$$j_{!*}\mathbb{L}\cong {\mathcal{G}}_{0}$$

of $G$-equivariant $\mathcal{D}(V)$-modules. The claim is presented as a natural expectation rather than a formally named conjecture, and the supplied text gives no resolution evidence; it remains open in the database.