Quasi-monomiality conjecture for valuations computing asymptotic Arnold multiplicities

Let $X$ be a smooth variety, let $\mathfrak{a}_{\bullet}$ be a graded sequence of ideals on $X$, and let $\mathfrak{q}$ be a nonzero ideal. Write $\operatorname{Arn}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})=\lambda^{-1}>0$, and let $\mathcal{J}(\mathfrak{a}_{\bullet}^{\lambda})$ denote the associated asymptotic multiplier ideal. A valuation $v\in\operatorname{Val}^*_X$ computes $\operatorname{Arn}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$ when it realizes the corresponding valuation formula.

Quasi-monomiality conjecture. The weak version asserts that, for every generic point $\xi$ of an irreducible component of the subscheme defined by $(\mathcal{J}(\mathfrak{a}_{\bullet}^{\lambda}):\mathfrak{q})$, there exists a quasi-monomial valuation $v\in\operatorname{Val}^*_X$ computing $\operatorname{Arn}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$ with $c_X(v)=\xi$. The strong version asserts that every valuation in $\operatorname{Val}^*_X$ computing $\operatorname{Arn}^{\mathfrak{q}}(\mathfrak{a}_{\bullet})$ is quasi-monomial.

The preceding theorem proves existence of a computing valuation with each specified generic center, but not quasi-monomiality. The source explicitly says that it cannot prove either conjectural version.