Thomas-Yau's minimizer conjecture for the Solomon functional

Let $\mathcal{L}$ be the class of admissible Lagrangian objects and let $\mathcal{S}$ be the Solomon functional. A minimizer is an element $L\in\mathcal{L}$ attaining the infimum of $\mathcal{S}$ on $\mathcal{L}$. Thomas-Yau minimizer conjecture. In the semistable case, the Solomon functional has a minimizer. This is one of the difficult implications in the variational program toward the Thomas-Yau existence conjecture; the source states that its proof would require several unproved compactness and smoothing assertions.