Herzog–Srinivasan upper multiplicity conjecture for arbitrary graded algebras

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring with its standard grading, let $I\subset S$ be a graded ideal, and let $R=S/I$ be a standard graded $K$-algebra. Write $c=\operatorname{codim}(R)$, and for each relevant homological degree define

$$M_i=\max\{j\in\mathbb Z:\beta^S_{i,j}(R)\neq0\},$$

where $\beta^S_{i,j}(R)=\dim_K\operatorname{Tor}^S_i(R,K)_j$ are the graded Betti numbers. Let $e(R)$ denote the multiplicity of $R$. Herzog–Srinivasan upper-bound conjecture. One has

$$e(R)\leq\frac{\prod_{i=1}^{c}M_i}{c!}.$$

This extends the upper multiplicity bound beyond the Cohen–Macaulay setting, where the two-sided multiplicity conjecture is stated; the source records the bound as conjectural after discussing cases in which the Cohen–Macaulay conjecture is known.