Huneke–Srinivasan multiplicity conjecture

Let $R=k[x_0,\dots,x_n]$, and let $J\subset R$ be a homogeneous ideal of codimension $c$ such that $R/J$ is Cohen–Macaulay. Let $e(R/J)$ denote the multiplicity of $R/J$. For the minimal graded free resolution of $R/J$, let $m_i$ be the minimal degree of a syzygy at step $i$, and let $M_i$ be the corresponding maximum.

Huneke–Srinivasan multiplicity conjecture. One has

$$\frac{1}{c!}\prod_{i=1}^c m_i\le e(R/J)\le\frac{1}{c!}\prod_{i=1}^c M_i.$$

This conjecture concerns sharp multiplicity bounds in terms of the degree shifts in a Cohen–Macaulay graded resolution. The supplied context says it is known in a number of special cases but remains open in general, even for monomial ideals in codimension three and above.