Maximal-time conjecture for the adapted Chern–Ricci flow

Let $(M,I,J,K,g)$ be a compact hyperhermitian manifold, let $\omega$ be its associated real $(1,1)$-form, and write $S^-:=\frac12(S-JS)$ for the $J$-anti-invariant part of a covariant $2$-tensor $S$. Consider the adapted Chern–Ricci flow

$$\dot\omega(t)=-\operatorname{Ric}^-(\omega(t)),\qquad \omega(0)=\omega.$$

Define

$$T=\sup\left\{t\geq 0:\,\text{there exists }\psi\in C^{\infty}(M)\text{ such that }\omega-t\operatorname{Ric}(\omega)^-+i(\partial\bar\partial\psi)^->0\right\}.$$

Maximal-time conjecture. There exists a unique maximal solution to the adapted Chern–Ricci flow on $[0,T)$.

This is proposed by analogy with the Kähler and Hermitian cases. The source provides no resolution evidence, so the conjecture remains open in this record.