Thomas's uniqueness conjecture for special Lagrangians under mean curvature flow

Let $L$ be a graded Lagrangian submanifold of a Calabi–Yau $n$-fold $X$, with the phase of the holomorphic volume form chosen so that the cohomological phase satisfies $\phi(L)=0$. Let $\theta$ denote the phase function of $L$. Assume either that, for every graded connect sum L_1\\#L_2\approx L, one has

$$[\phi(L_1),\phi(L_2)]\not\subseteq(\inf_L\theta,\sup_L\theta),$$

or that

$$\operatorname{vol}(L)\leq\int_{L_1}e^{-i\phi(L_1)}\Omega+\int_{L_2}e^{-i\phi(L_2)}\Omega$$

for all $L_1,L_2$ such that L\approx L_1\\#L_2. Thomas's conjecture. If $L$ satisfies either condition, then mean curvature flow for $L$ exists for all time and converges to a special Lagrangian in its Hamiltonian deformation class, namely the unique special Lagrangian conjectured by Thomas.