Rubinstein's Ricci iteration convergence conjecture

Let $X$ be a compact Kähler manifold with a Kähler class $\Omega$ containing a constant scalar curvature Kähler metric. For any $\omega\in\Omega$, consider the Ricci iteration

$$\frac{\omega_{i+1}-\omega_i}{\tau}=-\operatorname{Ric}(\omega_{i+1})+\operatorname{HRic}(\omega_{i+1}),\qquad i\in\mathbb{N},\qquad \omega_0=\omega.$$

Rubinstein's Ricci iteration convergence conjecture. The iteration exists for all $i\in\mathbb{N}$ and converges in an appropriate sense to a constant scalar curvature metric. The conjecture proposes a discrete uniformization procedure for cscK metrics. In the source paper, the authors state that they prove global existence and convergence modulo automorphisms, confirming Rubinstein's 2007 conjecture; the parser supplies no more specific resolution evidence.