Donaldson's coercivity conjecture for the K-energy

Let $(M,[\omega])$ be a polarized Kähler manifold, and let $E$ denote the K-energy functional. The maximal invariant, totally geodesic subspace induced by the automorphism group is the reference subspace for the relevant geodesic distance. Donaldson's coercivity conjecture. There exists a cscK metric if and only if the K-energy functional is coercive in terms of geodesic distance to the maximal invariant, totally geodesic subspace induced by the automorphism group. The preceding convexity conjecture is stated in the source as proved by Berman–Berndtsson and Chen–Li–Paun, but this coercivity equivalence is not resolved there.