Conjecture on extremal Betti numbers of Cartwright–Sturmfels ideals

Let $S$ be the polynomial ring with its ${\mathbb Z}^n$-grading, let $I$ be a Cartwright–Sturmfels (CS) ideal, and let $J$ be its ${\mathbb Z}^n$-graded generic initial ideal. The CS Betti-number conjecture. The extremal total Betti numbers of $I$ and $J$ are equal. In particular, $I$ and $J$ have the same projective dimension and Castelnuovo–Mumford regularity. This is motivated by the expected connection between the homological invariants of a CS ideal and those of its generic initial ideal; the source presents it as suggested by computational experiments, with no resolution stated.