Pu–Loewner inequality conjecture for the four-torus–projective-plane product

Assume the relative Pu inequality stated earlier. Let $\mathbf{g}$ be a Riemannian metric on $\mathbb T^2\times\mathbb R P^2$, and let $\operatorname{sys}\pi_1(\mathbf{g})$ and $\operatorname{stsys}_1(\mathbf{g})$ denote the relevant ordinary and stable 1-systoles.

Pu–Loewner inequality conjecture. Every such metric satisfies

$$\operatorname{sys}\pi_1(\mathbf{g})^2\operatorname{stsys}_1(\mathbf{g})^2\leq \frac{2}{\sqrt{3}}\frac{\pi}{2}\operatorname{vol}_4(\mathbf{g}),$$

while equality can occur only for a metric admitting a Riemannian submersion onto a Loewner-extremal torus with Pu-extremal fibers, namely real projective planes of constant Gaussian curvature. This is conditional on the relative Pu inequality, which the source identifies as the main obstacle to proving it.