The Chern-Yamabe flow curvature-bound conjecture

Let $X$ be the manifold underlying a Chern-Yamabe flow, and suppose the flow exists on

$$\Omega_T=X\times[0,T)$$

for some $T>0$. Denote its Chern scalar curvature by $S(x;t)$. Chern-Yamabe flow curvature-bound conjecture. There exists a constant $C(T)$ depending on $T$ such that

$$S(x;t)\leq C(T),\qquad \forall (x,t)\in\Omega_T.$$

Together with the preceding long-time existence criterion, this bound would fully resolve long-time existence of the Chern-Yamabe flow. The source presents it as a conjecture, but gives no resolution or evidence establishing its status.