Uniqueness and specialness conjecture for the minimizing valuation

Let $v_*$ be a minimizer of the valuation functional $\tilde{\beta}$ for a ${\mathbb{Q}}$-Fano variety $X$, and let $\mathcal{F}_{v_*}$ be the filtration induced by $v_*$. Minimizing valuation conjecture. The minimizer $v_*$ is unique and is special, meaning that $\mathcal{F}_{v_*}$ is a special ${\mathbb{R}}$-test configuration. This conjecture refines the existence result that a minimizer can be chosen among quasi-monomial valuations and predicts that the optimal valuation itself yields a special test configuration. The supplied text gives no resolution status.