Toric degeneration conjecture for Mori dream spaces

Let $X$ be a Mori dream space, and let $Y_\bullet$ be a flag for which the global semigroup $\Gamma_{Y_\bullet}(X)$ is defined. The Cox ring of $X$ is denoted by $\operatorname{Cox}(X)$, and $\operatorname{Spec}(\operatorname{Cox}(X))$ is its spectrum. Toric degeneration conjecture. Finite generation of $\Gamma_{Y_\bullet}(X)$ induces a degeneration of

$$\operatorname{Spec}(\operatorname{Cox}(X)),$$

which is compatible with the toric degenerations of $\Gamma_{Y_\bullet}(D)$ considered in the cited work. This would relate the global semigroup of a Mori dream space to toric degenerations of its Cox construction, extending the analogous phenomenon for semigroups associated with divisors.