Asymptotic regularity of quadratic linearly presented ideals

Determine whether every family of graded quadratic linearly presented ideals $I_n\subseteq k[x_1,\ldots,x_n]$ satisfies a finite upper bound for $\limsup_{n\to\infty}\operatorname{reg}(I_n)/\sqrt{n}$.

Background

The paper records an open question concerning the asymptotic Castelnuovo–Mumford regularity of arbitrary families of quadratic, linearly presented ideals in polynomial rings with an increasing number of variables. Affirmative answers were already known for quadratic monomial ideals and for defining ideals of Veronese and Grassmannian varieties.

The authors prove that the answer is affirmative for defining ideals of graded Möbius algebras associated to sequences of matroids. Their result does not resolve the question for general quadratic linearly presented ideals.