Vanishing conjecture for the chi-genus of Enriques moduli spaces

Let $Y$ be an Enriques surface and let $v=(r,\beta,n)\in H^{\ast}(Y,\mathbb{Z})$ be primitive, with $r$ even and $2\nmid\beta$. Let $M^Y(v)$ be the automatically smooth moduli space of stable sheaves on $Y$ with Chern character $v$ for a generic polarization. Vanishing conjecture for Enriques moduli spaces. The $\chi_t$-genus of $M^Y(v)$ vanishes:

$$\chi_{-t}(M^Y(v))=\sum_{p,q}(-1)^{p+q}h^{p,q}(M^Y(v))t^p=0.$$

The claim is presented as a consequence of the refined Vafa–Witten formula, but the author states that no proof is known. It concerns the Hodge-theoretic vanishing of moduli spaces of stable sheaves on Enriques surfaces under the stated parity conditions.