Brosnan's decomposition and Weyl-group monodromy conjecture

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$, let $M$ be a Hessenberg subspace, and let $\pi_M:G\times^B M\to\mathfrak{g}$ be the Hessenberg map. Let $\mathfrak{g}^{rs}$ be the Zariski open dense subset of regular semisimple elements, let $d=\dim(G\times^B M)$, and let $W$ be the Weyl group of $G$. For each local system $\mathcal{V}_i$ on $\mathfrak{g}^{rs}$, write $\operatorname{IC}(\overline{\mathfrak{g}^{rs}},\mathcal{V}_i)$ for its intermediate extension to $\mathfrak{g}=\overline{\mathfrak{g}^{rs}}$. Brosnan's conjecture. The complex $R\pi_{M*}\underline{\mathbb{C}}[d]$ is a direct sum of shifted intermediate extensions of irreducible local systems on $\mathfrak{g}^{rs}$, namely

$$R\pi_{M*}\underline{\mathbb{C}}[d]=\bigoplus \operatorname{IC}(\overline{\mathfrak{g}^{rs}},\mathcal{V}_i)[a_i],$$

where each $\mathcal{V}_i$ is irreducible and, for every $y\in\mathfrak{g}^{rs}$, the monodromy action of $\pi_1(\mathfrak{g}^{rs},y)$ on the stalk $\mathcal{V}_{i,y}$ factors through $W$. This conjecture concerns a decomposition-theorem description of the Hessenberg direct image together with Weyl-group control of its monodromy. The source gives no evidence of resolution, so its status remains open.