The filling pair systole lower-bound conjecture

Let $S_g$ be a closed orientable surface of genus $g$, let $\mathcal{M}_g$ denote its moduli space of hyperbolic metrics, and let $m_g$ be the minimum perimeter of a regular, right-angled $(8g-4)$-gon. Define $\mathcal{Y}_g:\mathcal{M}_g\rightarrow\mathbb{R}$ to be the filling pair systole function, which assigns to each hyperbolic metric the length of the shortest filling pair. Filling pair systole conjecture. For every hyperbolic metric in $\mathcal{M}_g$, one has

$$\mathcal{Y}_g\geq m_g/2.$$

This conjecture extends the lower bound established in the paper for filling pairs whose complement has at most two regions; it predicts the same universal lower bound for the shortest filling pair in general.