# Spines of Teichmuller space of a non-orientable surface

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?