Dimension conjecture for Thurston's equal-systole locus

Let $\mathcal{M}_g$ be the moduli space of closed hyperbolic surfaces of genus $g$, and let $\mathcal{X}_g$ denote Thurston's set of surfaces in $\mathcal{M}_g$ with equal-length systoles.

Dimension conjecture. For every $\varepsilon>0$, there exists an integer $g\geq 2$ such that $\mathcal{X}_g$ has dimension at least $(6-\varepsilon)g$.

The conjecture predicts that, for suitable genera, the equal-systole locus has dimension close to the full dimension $6g-6$ of moduli space. The paper establishes only that $\mathcal{X}_g$ has dimension at least $4g-5$ when $g$ is even, so the asserted asymptotic lower bound remains open.