Thomas-Yau's uniqueness conjecture for weak special Lagrangians

Let $X$ be a Calabi-Yau manifold, and let $L$ and $L'$ be special Lagrangian integral currents with the same phase angle $\hat{\theta}$. Equip them with suitable unobstructed brane structures and suppose that $L\simeq L'$ in $D^bFuk(X)$. Thomas-Yau uniqueness conjecture. Then $L=L'$ as integral currents. The claim extends the classical Thomas-Yau uniqueness expectation to a weak, possibly singular setting; the source gives a heuristic holomorphic-curve argument but explicitly notes that the argument is not completely rigorous.