Classical conjecture on the systolic extremality of the round real projective 3-space

Let $(\mathbb{R} \mathbb{P}^3,sph)$ denote real projective 3-space endowed with its metric of constant curvature. A metric is extremal for the isosystolic inequality if it realizes the optimal systolic constant in its topological manifold. Classical conjecture. The manifold $(\mathbb{R} \mathbb{P}^3,sph)$ is extremal for the isosystolic inequality. The round real projective 3-space is presented as a candidate for realizing the systolic constant of its topological manifold and is optimal in its conformal class. The conjecture concerns global, rather than merely conformal, systolic optimality; the supplied source does not state a resolution.