Wonderful polar resolution conjecture for orbit closures

Let $(\mathrm{G},V)$ be a prehomogeneous vector space, let $Z\subset V$ be the closure of an orbit of $\mathrm{G}$, and write $Z_{\mathrm{sing}}$ for its singular locus. For an integer $d\leq\dim V$ and a generic $d$-dimensional linear subspace $L\in\mathrm{G}(d,\dim V)$, let $\mathcal{P}(Z,L)$ denote the corresponding polar. Wonderful polar resolution conjecture. There exists an integer $d\leq\dim V$ such that, for a generic $L\in\mathrm{G}(d,\dim V)$, the polar $\mathcal{P}(Z,L)$ contains $Z_{\mathrm{sing}}$ and admits a wonderful resolution of singularities.

The conjecture is suggested by the preceding example of a polar whose successive blow-ups along its singular strata give a wonderful resolution. The source gives no further status information.