Yau's conjecture for the Hull–Strominger system

Let $X$ be a compact Calabi–Yau threefold with holomorphic volume form $\Omega$, and let $\mathfrak{b}_0$ be a balanced class on $X$. Let $V$ be a holomorphic vector bundle over $X$ satisfying

$$\deg_{\mathfrak{b}_0}(V)=0,\qquad \operatorname{ch}_2(V)=\operatorname{ch}_2(X)\in H^{2,2}_{BC}(X,\mathbb{R}).$$

Assume that $V$ is polystable with respect to $\mathfrak{b}_0$. Yau's conjecture. Then $(X,\Omega,V)$ admits a solution of the Hull–Strominger system. This conjecture asserts that the cohomological conditions, balancedness of the Calabi–Yau threefold, and polystability of the bundle are the only obstructions to solving the system; the existence and uniqueness problem remains widely open.