Continuity of the Seshadri constant at regular points of arbitrary codimension

Fix a model $X_\pi$ of $X$, a point $p\in X_\pi$ whose local ring $\mathcal{O}_{X_\pi,p}$ is regular of dimension $c\le \dim X$, and a regular system of parameters at $p$. Let $\varepsilon(D,\xi)$ denote the Seshadri constant associated with a divisor $D$ and a seminorm $\xi$. Continuity conjecture for the Seshadri constant. The continuity theorem for toric valuation spaces should also hold when $p$ is any such regular point, not only a smooth closed point. The supplied text explains that the existing proof using toric pairings does not cover this case because relevant curves need not have images meeting $p$ properly; thus the extension remains open.