Real analyticity conjecture for the volume function

Let $X$ be a smooth projective variety of dimension $d$, and let \textnormal{\Big}(X) denote its big cone. An open subset U\subseteq\textnormal{\Big}(X) is called “large” in the sense intended by the conjecture. Real analyticity conjecture. There is a “large” open set U\subseteq\textnormal{\Big}(X) such that $\textnormal{vol}_X$ is real analytic on each connected component of $U$. The volume function is known in general to satisfy homogeneity, continuity, and log-concavity, but its further regularity properties are largely unknown; this conjecture predicts real analyticity away from a suitably exceptional subset.