Eventual linear quotients for ideals with linear powers

Prove that if a (squarefree) monomial ideal I in the standard graded polynomial ring S = K[x1, …, xn] has linear powers, then the powers I^k have linear quotients for all sufficiently large integers k.

Background

The paper completely resolves when linear resolution coincides with linear powers for squarefree monomial ideals generated in degree d and when the class of such ideals is independent of the base field. In particular, for degrees d in {0,1,2,n−2,n−1,n} the authors show that a squarefree monomial ideal has a linear resolution if and only if all its powers have linear quotients, giving a strong uniform characterization.

They develop new criteria using x-dominant and y-dominant monomial orders to strengthen the classical x-condition on the Rees algebra: when the x-condition holds with respect to such orders, all powers have linear quotients. They then classify degree n−2 and 2 cases fully, and construct counterexamples in intermediate degrees showing that linear resolution does not imply linear powers in general.

Motivated by these results and existing literature, they formulate the conjecture that the linear powers property should force eventual linear quotients, positing a uniform asymptotic structural behavior for powers of monomial (in particular squarefree monomial) ideals with linear powers.