Sturmfels's regularity conjecture for unions of subspaces

Let $I$ be the vanishing ideal of a union of $d$ subspaces.

Sturmfels's conjecture. The Castelnuovo–Mumford regularity of $I$ is at most $d$.

This is a strengthening of the Subspace Conjecture, replacing a generator-degree bound by a bound on Castelnuovo–Mumford regularity. The source states that both conjectures were proven by the authors in the cited work.