Beheshti–Eisenbud containment conjecture for powers of general linear ideals

Let $R$ be a standard graded $k$-algebra of dimension $l+1$, and let $\mathfrak m$ be its maximal homogeneous ideal. Suppose that $R$ is a domain with isolated singularity, that $k$ is infinite, and that $I\subseteq R$ is generated by $l+1+c$ general linear forms. Set

$$\epsilon=\left\lfloor\frac{l}{c}\right\rfloor.$$

Beheshti–Eisenbud conjecture. For $q\gg0$,

$$\mathfrak m^{q+\epsilon}\subseteq I^q.$$

This is a conjecture about the asymptotic containment of powers of the homogeneous maximal ideal in powers of a general linear ideal. The paper proves a special case, while the full statement remains open in the source.