Quasimonomial valuation conjecture for graded sequences

Let $R$ be an excellent regular domain of equicharacteristic zero, let $\mathfrak a_\bullet$ be a graded sequence of ideals in $R$, and let $\mathfrak q$ be a nonzero ideal in $R$. A valuation $v$ computes $\operatorname{lct}^{\mathfrak q}(\mathfrak a_\bullet)$ when it achieves the infimum defining this jumping number. Quasimonomial valuation conjecture. There exists a quasimonomial valuation $v$ on $R$ that computes $\operatorname{lct}^{\mathfrak q}(\mathfrak a_\bullet)$.

The source notes that an arbitrary valuation computing the threshold is known to exist, and that the conjecture is true in dimensions one and two. The quasimonomial nature of the computing valuation remains open in higher dimensions.