Quasi-adjunction eigenspace formula for finite covers

Let $X$ be a smooth projective variety, let $D\subset X$ be a divisor, and let $G$ be a finite quotient of the fundamental group. Let $\chi\in\operatorname{Char}\pi_1(X-D)$ be a character of $G$, let $\bar U_G$ be a $G$-equivariant compactification of the unbranched cover of $X-D$ with Galois group $G$, and let $h^{n,0}_{\chi}(\bar U_G)$ denote the $\chi$-eigenspace dimension for the action of $G$ on $H^{n,0}(U_G)$. Let $\mathcal F$ be a contributing global face of quasi-adjunction, and let $D\in\operatorname{Pic}(X)$ be the divisor corresponding to the lift of $\chi$.

Quasi-adjunction eigenspace conjecture. Then $h^{n,0}_{\chi}(\bar U_G)=0$ unless the lift of $\chi$ belongs to a contributing global face of quasi-adjunction $\mathcal F$; in the latter case,

$$h^{n,0}_{\chi}(\bar U_G)={\rm \dim} H^1(\mathcal A_{\mathcal F}\otimes\Omega_X^n\otimes\mathcal O(D)).$$

The source presents this as a consequence of the preceding quasi-adjunction conjecture. Its notation depends on the referenced branched-cover lemma, and the supplied passage gives no independent resolution status.