Modified F-functional geodesic stability conjecture for Kähler–Ricci solitons

Let $(M,[\omega_0])$ be a Fano manifold, let ${\mathcal P}$ denote the space of Kähler potentials, and let ${\mathcal F}_X$ be the modified ${\mathcal F}$-functional associated with the vector field $X$. A geodesic ray is a path $\phi(t)$ in ${\mathcal P}$ satisfying the geodesic equation.

Modified F-functional geodesic stability conjecture. The following are equivalent:

1. There is a Kähler–Ricci soliton on $M$ with extremal vector field $X$.
2. There is a point $\phi\in{\mathcal P}$ such that, for any geodesic ray $\phi(t)$ starting at $\phi$, the derivative of the ${\mathcal F}_X$-functional is nonnegative for some $t_0\in[0,\infty)$.
3. For any geodesic ray $\phi(t)$, $t\in[0,\infty)$, the derivative of the ${\mathcal F}_X$-functional is nonnegative for some $t_0\in[0,\infty)$.

This conjecture is the Kähler–Ricci soliton analogue of the preceding geodesic stability formulation. The source connects the derivative of ${\mathcal F}_X$ with the modified Futaki invariant, but gives no resolution status.