The supernoetherianity conjecture for twisted homogeneous coordinate rings of abelian varieties

Let $X$ be an abelian variety and let $\sigma$ be an automorphism of $X$ that leaves invariant no proper subscheme. Let $\mathcal{L}$ be an ample invertible sheaf on $X$, and write $B(X,\mathcal{L},\sigma)$ for the associated twisted homogeneous coordinate ring. Supernoetherianity conjecture. The ring

$$B(X,\mathcal{L},\sigma)$$

is supernoetherian. This would provide higher-dimensional examples beyond the known supernoetherian algebras that are finite over commutative domains of Krull dimension at most one or covered by the paper's general theorem; the conjecture remains open.