Sufficiently general arcs have minimal valuation above the Nagata threshold

Let $X=\mathbb P^2$ with a fixed point $O$, and let $C$ be an arc in the valuative tree $\mathcal{QM}$ of quasi-monomial valuations centred at $O$. Write $\widehat\mu(C,s)$ for the asymptotic multiplicity invariant associated with the valuation indexed by $s\in[1,+\infty)$. Here, $C$ is understood to be sufficiently general in the sense specified later in the source.

Minimal valuation conjecture. If $C$ is sufficiently general and $s\geqslant 8+\frac{1}{36}$, then

$$\widehat\mu(C,s)=\sqrt{s}.$$

This is expected to follow from the absence of supraminimal curves along sufficiently general arcs, and it would imply Nagata's conjecture that the inverse of the $t$-point Seshadri constant of $\mathbb P^2$ equals $\sqrt t$ for $t\geqslant9$.