Mendez–Pinto–Villarreal conjecture for Simis monomial ideals

Let $S$ be a polynomial ring and let $I$ be a monomial ideal in $S$. An ideal is Simis when its symbolic Rees algebra is generated by its elements of degree one; a monomial ideal has a minimal irreducible decomposition when its irreducible decomposition is minimal, and a standard linear weighting $w$ assigns positive integer weights to the variables in the standard way, producing the weighted ideal $J_w$. Assume that $I$ has no embedded associated primes, that its irreducible decomposition is minimal, and that $I$ is a Simis ideal. Mendez–Pinto–Villarreal conjecture. There exists a Simis square-free monomial ideal $J$ and a standard linear weighting $w$ such that

$$I=J_w.$$

The conjecture proposes that these Simis monomial ideals arise precisely by applying standard linear weightings to Simis square-free monomial ideals. It is presented here as a recent conjecture and no resolution is supplied in the source.