Converse conjecture to the dHYM stability inequalities for Kähler threefolds

Let $(X,\omega)$ be a Kähler threefold and let $L\rightarrow X$ be a holomorphic line bundle. Suppose that the Chern number inequality holds, the lifted angle is well-defined with $\hat{\theta}\in(\pi,\frac{3\pi}{2})$, $Z_X(L)$ lies in the upper half-plane, and for every irreducible analytic subset $V\subset X$ one has $Z_V(L)$ in the upper half-plane and $\varphi_V(L)>\varphi_X(L)$. C.-Yau converse conjecture. The converse of the stated proposition holds: these numerical and slicing-angle conditions imply that $L$ admits a solution of the deformed Hermitian–Yang–Mills equation with lifted angle $\hat{\theta}\in(\pi,\frac{3\pi}{2})$. The source notes that the small-radius-limit case for ample line bundles on toric varieties, and later the corresponding result without the toric assumption, had been proved; the general converse was presented as the remaining conjectural statement.