Kalai's monomial-ideal diameter conjecture

Let $I\subset k[x_1,\dots,x_n]$ be a square-free monomial ideal generated in degree $d$ and having a linear presentation. Let $G_I$ be the graph whose vertices are the minimal monomial generators of $I$, labeled by the indices of their factors. Kalai's monomial-ideal diameter conjecture. The diameter of $G_I$ is bounded above by a polynomial in $d$. This is the algebraic restatement of Kalai's Abstract Polynomial Hirsch Conjecture, connecting graph diameter bounds with linear presentations of square-free monomial ideals; the source does not specify whether the conjecture has been resolved.