Almost Cohen–Macaulayness of Rees algebras of monomial almost complete intersections

Let $I$ be a monomial ideal of $k[x_1,\ldots,x_n]$ of finite colength. If $I$ is an almost complete intersection, then its Rees algebra ${\bf R}[It]$ is almost Cohen–Macaulay, where ${\bf R}=k[x_1,\ldots,x_n]$.

Monomial Rees-algebra conjecture.

$${\bf R}[It]\text{ is almost Cohen--Macaulay}.$$

The source proves particular examples and proposes the assertion for all finite-colength monomial almost complete intersections; it also asks separate questions about Gorenstein ideals.