Pu's relative 1-systole conjecture

Let $S$ be a nonorientable surface and let $\phi:\pi_1(S)\to \mathbb Z_2$ correspond to a map $\hat\phi:S\to \mathbb R P^2$ of absolute degree one. For a metric $\mathbf{g}$ on $S$, define the relative 1-systole by

$$\phi\operatorname{sys}_1(\mathbf{g})=\min_{\phi([\gamma])\ne 0\in\mathbb Z_2}\operatorname{length}(\gamma).$$

Pu's relative 1-systole conjecture. For any such $S$ and $\hat\phi$,

$$\phi\operatorname{sys}_1(\mathbf{g})^2\leq \frac{\pi}{2}\operatorname{area}(\mathbf{g}).$$

This is a sharp generalization of Pu's inequality, with equality expected to reflect the projective-plane constant-curvature case; the source gives no resolution of the conjecture.