Weight-filtration conjecture for spectral-cover eigenspaces

Let $\rho\colon G\to GL(V)$ be an irreducible representation, let $\lambda$ be a dominant weight, let $W_\lambda$ be its stabilizer in $W$, and let $m_\lambda$ be the multiplicity of $\lambda$ in $\rho$. For each dominant weight $\lambda$, define

$$P_\lambda=\prod_{\lambda'\in W\cdot\lambda}(\rho_*\overline\Sigma-\lambda'\cdot\operatorname{Id}),$$ $$Q_\lambda=\prod_{\lambda'\leq\lambda}P_{\lambda'},\qquad Q_\lambda^0=\prod_{\lambda'<\lambda}P_{\lambda'},$$

and set $F_\lambda=\operatorname{Ker}Q_\lambda$ and $F_\lambda^0=\operatorname{Ker}Q_\lambda^0$ inside $V\otimes_\mathbb{C}S^W$. Weight-filtration conjecture. There is an isomorphism of $S^W$-modules

$$\hat h\colon F_\lambda/F_\lambda^0\cong (S^{W_\lambda})^{m_\lambda},$$

and it satisfies

$$\hat h\circ\rho_*\overline\Sigma=\lambda\hat h.$$

The construction decomposes the spectral-cover module according to dominant weights and their partial ordering; the source presents this expected description of each successive piece as a conjecture and gives no resolution.