The extended multiplicity conjecture for Cohen–Macaulay graded algebras

Let $R=k[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field, let $I\subset R$ be a graded ideal, and set $c=\operatorname{codim} R/I$. For a minimal graded free resolution of $R/I$, let $m_i$ and $M_i$ denote the minimum and maximum shifts in homological degree $i$. Assume that $R/I$ is Cohen–Macaulay. Extended multiplicity conjecture. One has

$$\frac{1}{c!}\prod_{i=1}^c m_i\leq e(R/I)\leq\frac{1}{c!}\prod_{i=1}^c M_i,$$

and equality in the lower bound, respectively the upper bound, holds if and only if $R/I$ has a pure resolution. This extends the original multiplicity conjecture by including the characterization of equality; the statement was known in the cited work for Cohen–Macaulay algebras of codimension two and Gorenstein algebras of codimension three, while the general case remains open.