Vanishing of top equivariant Chern classes of regular embeddings

Let $G$ be a connected affine algebraic group and $X$ a regular $G$-variety (in the sense of Bifet-De Concini-Procesi) with open orbit $G/H$ and boundary divisor $D$. We show the vanishing of the $G$-equivariant Chern classes of the bundle of differential forms on $X$ with logarithmic poles along $D$, in degrees larger than $\dim(X) - \rk(G) + \rk(H)$. Our motivation comes from Gieseker's degeneration method to prove the Newstead-Ramanan conjecture on the vanishing of the top Chern classes of the moduli space of stable vector bundles on a curve.