Continuity conjecture on higher rank skeleta

Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $\kappa$, and let $L$ be a big line bundle on $X$. For each valuation $\nu$ in the higher rank valuation space $X^{\mathrm{bir},d}$, let $\Delta_\nu$ be the corresponding Newton–Okounkov body, giving a map $\Delta\colon X^{\mathrm{bir},d}\to\mathrm{BC}(\mathbb{R}^d)$. Higher rank skeleton continuity conjecture. The restriction of $\Delta$ to each higher rank skeleton is continuous. This is a heuristic formulation of the continuity expected for Newton–Okounkov bodies along higher rank skeleta; the supplied text gives no proof or resolution.