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 $M_{\mathcal{A}\mathrm{/}Y}^H(2,c_1,{\mathrm{ch}}_2)$ with virtual dimension $N$. We show that the virtual Euler characteristic $e^{\mathrm{v}\mathrm{i}\mathrm{r}}(M)$ only depends on $N$, more precisely, $e^{\mathrm{v}\mathrm{i}\mathrm{r}}(M)=0$ when $N$ is odd and $e^{\mathrm{v}\mathrm{i}\mathrm{r}}(M)=2\cdot e(Y^{[\frac{N}{2}]})$ when $N$ is even.