Relative Pu conjecture for orientable surfaces with an involution

Let $X$ be an orientable surface of even genus with a Riemannian metric $\mathcal G$ that admits a fixed-point-free, orientation-reversing, isometric involution $\tau$. For $p\in X$, write $\operatorname{dist}_{\mathcal G}(p,\tau(p))$ for the distance between $p$ and its image, and write $\operatorname{area}(\mathcal G)$ for the area of the metric.

Relative Pu conjecture. There is a point $p\in X$ such that

$$\frac{\operatorname{dist}_{\mathcal G}(p,\tau(p))^2}{\operatorname{area}(\mathcal G)}\leq\frac{\pi}{4}.$$

This is a relative version of Pu's inequality arising from fillings of the circle by positive-genus surfaces. The source does not provide evidence of a resolution; it later proves the corresponding result for genus one fillings.