Thomas-Yau's equivalent variational existence conjecture

Let $\mathcal{L}$ be the class of quantitatively almost calibrated exact unobstructed Lagrangian objects under consideration, and let $L_0\in\mathcal{L}$ be exact, quantitatively almost calibrated, and unobstructed. Let $\mathcal{S}$ denote the Solomon functional, and let a destabilizing distinguished triangle mean a distinguished triangle in $\mathcal{L}$ satisfying the Thomas-Yau destabilizing phase condition. Thomas-Yau's variational conjecture. Assuming Thomas-Yau semistability for $L_0$, the following statements are equivalent: $\mathcal{L}$ contains a special Lagrangian representative; there is no destabilizing distinguished triangle in $\mathcal{L}$; $\mathcal{S}$ is bounded below on $\mathcal{L}$; and $\mathcal{S}$ has a minimizer in $\mathcal{L}$. The source presents these equivalences as the variational form of the Thomas-Yau existence conjecture, with several implications supported only under additional hypotheses and the remaining directions unproved.