The minimal-discrepancy Nash valuation conjecture for 3-fold terminal singularities

Let $(X,p)$ be a 3-fold terminal singularity with Gorenstein index $m\geq 1$. An exceptional prime divisor $E$ over $X$ has discrepancy $a(E,X)$, and its associated valuation is denoted by $\operatorname{val}_{E}$. A Nash valuation is a valuation arising from an irreducible component of the space of arcs through the singular point. The minimal-discrepancy Nash valuation conjecture. Every exceptional prime divisor $E$ over $X$ with minimal discrepancy, namely $a(E,X)=\frac{1}{m}$, induces a Nash valuation $\operatorname{val}_{E}$ of $X$. This is the weaker version proposed alongside the discrepancy-at-most-one conjecture and is intended to characterize Nash valuations associated with minimal discrepancy over general 3-fold terminal singularities.