Migliore-Nagel-Römer equality characterization conjecture

Let $V$ be a finite set, let $R=\Bbbk[V]$ be a standard graded polynomial ring, and let $I$ be a homogeneous ideal of height $c$. Let $e(R/I)$ be the Hilbert-Samuel multiplicity, and let $M_l$ and $m_l$ denote respectively the largest and smallest twists in homological degree $l$ of a minimal graded free resolution of $R/I$. A graded resolution is pure when $M_l=m_l$ for every $l$. Migliore-Nagel-Römer's equality characterization conjecture. If equality holds in either

$$e(R/I)=\frac{M_1M_2\cdots M_c}{c!}$$

or, assuming $R/I$ is Cohen–Macaulay,

$$e(R/I)=\frac{m_1m_2\cdots m_c}{c!},$$

then $R/I$ is Cohen–Macaulay with a pure resolution. The source reports that this implication is known when $R/I$ is Cohen–Macaulay and has a quasi-pure resolution, but does not resolve it in general.