Herzog-Srinivasan Taylor multiplicity bound conjecture

Let $V$ be a finite set, let $R=\Bbbk[V]$ be a standard graded polynomial ring, and let $I$ be a monomial ideal of height $c$. Let $e(R/I)$ denote the Hilbert-Samuel multiplicity. If $\mathbb T_\bullet$ is the Taylor resolution of $R/I$, let $T_l$ be the largest twist in homological degree $l$; equivalently,

$$T_l=\max\{\deg\operatorname{lcm}(f_{s_1},\ldots,f_{s_l}):1\leq s_1<\cdots<s_l\leq m\}.$$

Herzog-Srinivasan's Taylor multiplicity bound conjecture.

$$e(R/I)\leq\frac{T_1T_2\cdots T_c}{c!}.$$

The paper studies this weaker Taylor bound for quadratic monomial ideals and states that it is proved there for all such ideals; the supplied text does not establish the claim for arbitrary monomial ideals.