Ustinovskiy's convergence conjecture for left-invariant HCF+ solutions

Let $\mathsf{G}$ be a simple complex Lie group. A left-invariant solution is a solution of the Hermitian curvature flow equation that is invariant under left translations, and $g_{\operatorname{can}}$ denotes the canonical left-invariant metric on $\mathsf{G}$. Ustinovskiy's convergence conjecture. After some rescaling, any left-invariant solution on $\mathsf{G}$ converges in the $C^\infty$ topology to $\phi^*g_{\operatorname{can}}$ for some inner automorphism $\phi$ of $\mathsf{G}$. This conjecture concerns the asymptotic behaviour of Hermitian curvature flow on simple complex Lie groups and is presented as an open conjecture in the supplied text.