Continuity conjecture for variations of Newton–Okounkov bodies

Let $X$ be a projective variety of dimension $d$, let $L$ be a big line bundle on $X$, and let $D$ be a simple normal crossing divisor whose dual cone complex $\Sigma(D)$ is pure of dimension $d$. The tangent cone $T\mathcal{C}^{d-1}(\Sigma(D))$ parametrizes the relevant higher rank valuations, and the associated Newton–Okounkov bodies define a map to the space of compact subsets of $\mathbb{R}^d$ with the Hausdorff distance. Continuity conjecture. The variation of Newton–Okounkov bodies on $T\mathcal{C}^{d-1}(\Sigma(D))$ is continuous. This is presented as the objective of the paper’s study; the supplied text gives no proof or resolution, so the conjecture remains open.