Killing-form convergence conjecture for Hermitian curvature flow on complex simple groups

Let $G$ be a complex Lie group whose Lie algebra is the complexification $\mathfrak g=\mathfrak g_{\mathbb R}\otimes\mathbb C$ of a simple compact real Lie algebra $\mathfrak g_{\mathbb R}$ with invariant metric $\langle\,\ \rangle$. Let $\kappa\in\operatorname{Sym}^{1,1}(\mathfrak g)$ correspond to $\langle\,\ \rangle^{-1}\in\operatorname{Sym}^{2}(\mathfrak g_{\mathbb R})$; this element is called the Killing form. Let $h(t)$ solve the Hermitian curvature flow ODE on its maximal time interval $[0,t_{\max})$. Killing-form convergence conjecture. There exist $g\in G$ and $\lambda\in\mathbb R$ such that $h(t)$ pinches towards $\lambda\operatorname{Ad}_g(\kappa)$, namely

The conjecture describes the expected asymptotic behavior of the reduced Hermitian curvature flow on complex Lie groups with simple compact real form: after normalization, the solution should converge to a translate and scalar multiple of the Killing form. The supplied text gives no resolution, so its status remains open.