Strong integrality conjecture for weakly symmetric representations

Let $G$ be a reductive group with maximal torus $T$, let $V$ be a weakly symmetric representation of $G$, and let $\lambda\in X_*(T)$. Write $G^{\lambda}$ for the centralizer of the one-parameter subgroup $\lambda$, $G_{\lambda}$ for the subgroup acting trivially on the relevant fixed-point subspace, and $V^{\lambda}$ for the $\lambda$-fixed subspace. Let $\mathcal{P}_{\lambda}$ denote the corresponding cohomological integrality space, and let $\mathrm{IH}^*$ denote intersection cohomology. Strong integrality conjecture. There is a canonical identification

$$\mathcal{P}_{\lambda}\cong \mathrm{IH}^*(V^{\lambda}/\\!/G^{\lambda})$$

if a general closed orbit of $G^{\lambda}/G_{\lambda}$ inside $V^{\lambda}$ has finite stabilizer in $G^{\lambda}/G_{\lambda}$, and $\mathcal{P}_{\lambda}=0$ otherwise. This proposed sheafified strengthening of cohomological integrality would provide an algorithm for computing the intersection cohomology of GIT quotients for symmetric representations of reductive groups; it is the subject of forthcoming work, so its resolution is not established here.