Strict convexity conjecture for the Mabuchi functional

Let ${\mathcal M}$ be the Mabuchi functional and let ${\mathcal G}$ be a $C^{1,1}$-geodesic segment. Strict convexity conjecture. If ${\mathcal M}$ is linear along ${\mathcal G}$, then ${\mathcal G}$ is generated by a holomorphic vector field. This is the rigidity statement underlying strict convexity of the Mabuchi functional: equality in its convexity should arise only from holomorphic symmetries. The source presents this as the question investigated after deriving a pointwise formula, and no resolution is given here.