Chen's convexity conjecture for the K-energy

Let $u_0,u_1\in\mathcal H$, and let $\{u_t\}_{t\in[0,1]}$ be the $C^{1,1}$ geodesic connecting them.

Chen's convexity conjecture. The function

$$t\longmapsto K(u_t)$$

is convex on $[0,1]$.

Convexity of the $K$-energy along geodesics is central to the variational approach to cscK metrics. The source presents this as a conjecture after recalling convexity of the $J_\chi$ functional; no resolution is stated.