Chern–Schwartz–MacPherson class conjecture for locally quasi-homogeneous free divisors

Let $X$ be a locally quasi-homogeneous free divisor in a nonsingular variety $V$. Write $c_{\mathrm{SM}}(1_{V\smallsetminus X})$ for the Chern–Schwartz–MacPherson class of the complement, and let $\Omega^1_V(\log X)$ denote the sheaf of differential forms with logarithmic poles along $X$. Chern–Schwartz–MacPherson class conjecture. One has

$$c_{\mathrm{SM}}(1_{V\smallsetminus X})=c\bigl(\Omega^1_V(\log X)^\vee\bigr)\cap[V].$$

This extends the corresponding equality proved in the paper for complements of hyperplane arrangements whose associated affine arrangements are free; the conjecture is proposed as consistent with all known cases, but its general status is not specified in the source.