Euler characteristics of moduli of twisted sheaves on Enriques surfaces

Let $Y$ be an Enriques surface and let $\mathcal{A}$ be an Azumaya algebra corresponding to the non-trivial Brauer class. Let $M$ be the moduli space of stable twisted sheaves on Enriques surfaces with twisted Chern character $M^H_{\mathcal{A}/Y}(2,c_1,\operatorname{ch}_2)$ with virtual dimension $N$. We show that the virtual Euler characteristic $e^\mathrm{vir}(M)$ only depends on $N$, more precisely, $e^\mathrm{vir}(M)=0$ when $N$ is odd and $e^\mathrm{vir}(M)=2\cdot e(Y^{[\frac{N}{2}]})$ when $N$ is even.