Smooth convergence conjecture for the Chern-Ricci flow on elliptic bundles

Let $f:M\to\Sigma$ be an elliptic bundle over a Riemann surface $\Sigma$ of genus at least $2$, and let $\omega(t)$ solve the Chern-Ricci flow on $M$ from an arbitrary Gauduchon metric $\omega_0$. Let $\omega_\Sigma$ be the Kähler-Einstein metric on $\Sigma$, and let $d_\Sigma$ be its induced distance. Elliptic-bundle convergence conjecture. As $t\to\infty$,

$$\frac{\omega(t)}{t}\to f^*\omega_\Sigma$$

smoothly on $M$, $(M,\omega(t)/t)\to(\Sigma,d_\Sigma)$ in the Gromov-Hausdorff sense, and there is a constant $C$ such that

$$\sup_M\left|\mathrm{Rm}\left(\frac{\omega(t)}{t}\right)\right|_{g(t)/t}\leqslant C$$

for all sufficiently large $t$. The same conclusions are expected for a general non-Kähler minimal properly elliptic surface after passing to a finite cover making the fibration elliptic. These claims extend the known convergence results for special initial metrics.