Let $S_{g,n}$ be the orientable surface of genus $g$ and $n$ punctures. Denote $\Gamma_{g,n}$ be the mapping class group of $S_{g,n}$ and $\mathcal T_{g,n}$ the Teichmüller space of $S_{g,n}$.

In http://www.math.lsa.umich.edu/~lji/a-tale-of-two-groups.pdf page 34 item (17) it is mentioned that, provided $n>0$, $\mathcal T_{g,n}$ has a $\Gamma_{g,n}$-equivariant spine of dimension equal to the virtual cohomological dimension of $\Gamma_{g,n}$.

Is there an analogous result for a non-orientable surface with at least one puncture? That is, is it true that the Teichmüller space of a non-orientable surface with at least one puncture has a spine of dimension equal to the virtual cohomological dimension of the corresponding mapping class group?