Long-time convergence conjecture for generalized Kähler-Ricci flow on Fano varieties

Suppose $(g, I, J)$ is a symplectic type generalized Kähler structure on a Fano manifold $(M, J)$, with $F \in 2\pi c_1(M, J)$. Let $(g_t,I_t,J)$ be the solution of the normalized generalized Kähler-Ricci flow with this initial data. Generalized Kähler-Ricci flow convergence conjecture. The solution exists for all $t\in[0,\infty)$. If $(M,J)$ admits a Kähler-Einstein metric, then the solution converges in the $C^{\infty}(M)$ topology to a Kähler-Einstein metric. If $(M,J)$ admits a Kähler-Ricci soliton with soliton vector field $K$, and $(g,I,J)$ is invariant under the torus action generated by $K$, then there is a smooth family of complex automorphisms $\tau_t$ of $(M,J)$ such that the pullback of $g_t$ by $\tau_t$ converges in the $C^{\infty}(M)$ topology to a Kähler-Ricci soliton metric in $c_1(M,J)$ with soliton vector field $K$. If $(M,J)$ is a toric Fano variety and $(g,I,J)$ is invariant under the maximal torus $\mathbb T$, then the reduced equation has a global solution $\phi_t(y)$ on $[0,\infty)\times\mathbb R^m$, and there are points $y_t\in\mathbb R^n$ and real constants $c_t$ such that

$$\widetilde\phi_t(y):=\phi_t(y+y_t)+c_t$$

converges in $C^{\infty}([0,\infty)\times\mathbb R^m)$ to a smooth convex function $\widetilde\phi_{\infty}(y)$ on $\mathbb R^m$ defining a $\mathbb T$-invariant Kähler-Ricci soliton on $M$. This conjecture is a generalized Kähler analogue of the corresponding long-time existence and convergence results for normalized Kähler-Ricci flow on Fano manifolds. The nonsingular convergence theorem and the toric scalar convergence results provide partial evidence, but the full assertion remains open.