On the dimension of the mapping class groups of a non-orientable surface

Let $\mathcal{N}_g$ be the mapping class group of a non-orientable closed surface. We prove that the proper cohomological dimension, the proper geometric dimension, and the virtual cohomological dimension of $\mathcal{N}_g$ are equal whenever $g\neq 4,5$. In particular, there exists a model for the classifying space of $\mathcal{N}_g$ for proper actions of dimension $\mathrm{vcd}(\mathcal{N}_g)=2g-5$. Similar results are obtained for the mapping class group of a non-orientable surface with boundaries and possibly punctures, and for the pure mapping class group of a non-orientable surface with punctures and without boundaries.
Comment: 20 pages. Version accepted for publication in Homology, Homotopy and Applications