Uniform equivalence conjecture for compatible Calabi–Yau metrics

Let $X$ be a normal affine variety, let $v$ be a fixed polystable negative valuation on $X$, and let $\mathcal M_v$ be the space of compatible Calabi–Yau metrics $\omega$ on $X$ satisfying $v=v_\omega$. Uniform equivalence conjecture. If $\mathcal M_v$ is nonempty, then there exists a constant $C>0$ such that for all $\omega_1,\omega_2\in\mathcal M_v$,

$$C^{-1}\omega_1\leqslant\omega_2\leqslant C\omega_1.$$

Uniform equivalence is necessary for two compatible Calabi–Yau metrics to have the same valuation, and the analogous local question for Kähler–Einstein metrics on varieties with klt singularities is also open. The conjecture itself remains open.