The pure Poincaré polynomial identity for quiver and multiplicative varieties

Let ${\bf \Sigma}={\bf \sigma}{\bf C}$, with ${\bf C}$ a unipotent conjugacy class of ${\bf L}$ and $\sigma\in Z_{\bf L}$. Put ${\bf C}'={\bf C}-1$, and assume that there exists $\sigma'\in{\mathcal{A}}_{({\bf L,C}')}$ such that $C_{\GL_n}(\sigma)=C_{\GL_n}(\sigma')$. Let $\mathbf{w}\in W({\bf L,\Sigma})$; denote by $PH_c^{\bf w}$ and $P_c^{\bf w}$ the indicated pure and ordinary Poincaré polynomials of $\mathbb{M}_{\bf L,P,\Sigma}$ and $\mathbb{Q}_{\bf L,P,\Sigma'}$, respectively.

Quiver–multiplicative Poincaré polynomial conjecture.

$$PH_c^{\bf w}\left(\mathbb{M}_{\bf L,P,\Sigma},q\right)=P_c^{\bf w}\left(\mathbb{Q}_{\bf L,P,\Sigma'},q\right).$$

In the semisimple case with ${\bf w}=1$, this conjecture is attributed to T. Hausel; the statement is open in the generality given here.