The dimension of Thurston's spine

We show that for every $\epsilon >0$, there exists some $g\ge 2$ such that the set of closed hyperbolic surfaces of genus $g$ whose systoles fill has dimension at least $(5-\epsilon )g$. In particular, the dimension of this set – proposed as a spine for moduli space by Thurston – is larger than the virtual cohomological dimension of the mapping class group.