Thomas-Yau's Harder-Narasimhan obstruction conjecture

Let $X$ be a Calabi-Yau manifold with Kähler metric. Let $L$ be an almost calibrated exact Lagrangian brane in $D^bFuk(X)$ with a Harder-Narasimhan decomposition

$$0=\mathcal{E}_0\to\mathcal{E}_1\to\cdots\to\mathcal{E}_N=L,$$

whose distinguished triangles are

$$\mathcal{E}_{i-1}\to\mathcal{E}_i\to L_i\to\mathcal{E}_{i-1}[1],$$

with $L_i\in\mathcal{P}(\phi_i)$ and $\phi_1>\cdots>\phi_N$. Put $\hat{\theta}_i=\pi\phi_i=\arg\int_{L_i}\Omega$. Thomas-Yau's Harder-Narasimhan conjecture. The phase angle inequality and the volume lower bound associated with this decomposition hold. These inequalities are intended to provide Floer-theoretic obstructions to special Lagrangian representatives and to connect Joyce's Harder-Narasimhan picture with Lagrangian geometry; the assertion is presented as conjectural in the source.