Conjecture on Kähler-Ricci flow near deformed complex structures

Let $(M,J_0)$ be a Fano manifold admitting a Kähler-Ricci soliton $\omega_{KS}\in 2\pi c_1(M,J_0)$. Let $H_1$ be the operator whose negative eigenspaces define the linear subspace $A_-\subset\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)$, and let $B(\epsilon)$ be the parameter ball, with $\psi_\tau$ the deformation associated to $\tau$ and $Z$ the previously defined subspace. The Kähler-Ricci flow deformation conjecture. There exists a sufficiently small $\epsilon>0$ such that: (1) if $\tau\in B(\epsilon)$ and $\psi_\tau\in A_-\cup Z$, then the Kähler-Ricci flow converges smoothly, for every initial metric $\omega_0\in 2\pi c_1(M,J_{\psi_\tau})$, to a Kähler-Ricci soliton $(M,J_\infty,\omega_\infty)$ at polynomial rate, with $\lambda(\omega_\infty)=\lambda(\omega_{KS})$; and (2) if $\tau\in B(\epsilon)$ and $\psi_\tau\in\mathcal{H}^{0,1}_{\theta}(M,T^{1,0}M)\setminus(A_-\cup Z)$, then the flow converges, for every initial metric $\omega_0\in 2\pi c_1(M,J_{\psi_\tau})$, to a singular Kähler-Ricci soliton $\omega_\infty$ with $\lambda(\omega_\infty)>\lambda(\omega_{KS})$. This conjecture describes the expected behavior of the Kähler-Ricci flow under small deformations of the complex structure: the stable deformation directions should lead to smooth solitons with unchanged entropy, whereas the remaining directions are expected to produce singular solitons with strictly larger entropy. The preceding theorem establishes the smooth convergence conclusion under the stronger assumption that every nearby deformation admits a Kähler-Ricci soliton and that $H_1\leq 0$ on the relevant harmonic space; the two-regime assertion above is presented as a conjecture.