Affine-space Abhyankar valuation conjecture for log canonical thresholds

Let $X=\mathbb{A}^n_k$, where $\mathbb{F}_p\subset k=\overline{k}$, and let $\mathfrak{a}_\star$ be a graded sequence of ideals on $X$ with $\operatorname{lct}(\mathfrak{a}_\star)<\infty$, vanishing only at a closed point $x\in X$.

Affine-space Abhyankar valuation conjecture. The following versions should hold:

• Weak version: some Abhyankar valuation centered at $x$ computes $\operatorname{lct}(\mathfrak{a}_\star)$.
• Strong version: every valuation of transcendence degree $0$ over $\mathbb{A}^n_k$, centered at $x$, and computing $\operatorname{lct}(\mathfrak{a}_\star)$ is Abhyankar.

This is the affine-space reduction proposed as a more approachable form of the preceding conjecture. The source gives no resolution.