Thomas-Yau's minimizer-to-special-Lagrangian conjecture

Let $\mathcal{L}$ be the admissible class of Lagrangian currents and let $\mathcal{S}$ be the Solomon functional. Let $\hat{\theta}$ be the phase determined by the relevant Fukaya-category class. A minimizer is an element of $\mathcal{L}$ attaining the minimum of $\mathcal{S}$. Thomas-Yau's minimizer conjecture. Every minimizer $L$ of the Solomon functional inside $\mathcal{L}$ is a special Lagrangian of phase $\hat{\theta}$. This is the variational step converting existence of a minimizer into existence of a special Lagrangian; the source identifies regularity and admissible-variation issues as major gaps.