Tsai–Tsui–Wang area-nonincreasing rigidity conjecture

Let $(\mathbb S^n,g_{std})$ and $(\mathbb{S}^m,g_{std})$ be round spheres, and let $(\mathbb C \mathbb P ^n,g_{FS})$ and $(\mathbb C \mathbb P ^m,g_{FS})$ be complex projective spaces with the Fubini–Study metrics. Consider an area-nonincreasing map between either pair of spaces. Tsai–Tsui–Wang's conjecture. Any area-nonincreasing map between $(\mathbb S^n,g_{std})$ and $(\mathbb{S}^m,g_{std})$ or between $(\mathbb C \mathbb P ^n,g_{FS})$ and $(\mathbb C \mathbb P ^m,g_{FS})$ must be either homotopically trivial or an isometry. The conjecture extends rigidity from distance-nonincreasing maps to the weaker area-nonincreasing condition; the source says it was speculated by Tsai, Tsui, and Wang and does not state a resolution.