Polynomial-ring quasimonomial valuation conjecture for graded sequences

Let $k$ be an algebraically closed field of characteristic zero and let $R=k[x_1,\ldots,x_m]$. Let $\mathfrak q$ be a nonzero ideal and let $\mathfrak a_\bullet$ be a graded sequence of ideals in $R$. Polynomial-ring quasimonomial valuation conjecture. If $\mathfrak a_1\supseteq\mathfrak m^p$ for some maximal ideal $\mathfrak m$ of $R$ and some $p\ge1$, then there is a quasimonomial valuation $v$ on $R$ that computes $\operatorname{lct}^{\mathfrak q}(\mathfrak a_\bullet)$.

This is presented as a special case of the preceding conjecture. The source states that it is true in dimensions one and two, with the higher-dimensional case left open.