Moreno-Socias's almost-degrevlex conjecture for generic ideals

Let $I$ be an ideal generated by generic forms in $k[x_1,\ldots,x_n]$, and take its initial monomial ideal in degrevlex order. A monomial ideal is almost degrevlex if $m\in I$ implies that every monomial $m'$ larger than $m$ with $\deg m'=\deg m$ also lies in $I$.

Moreno-Socias's conjecture. The initial monomial ideal in degrevlex order of an ideal generated by generic forms is almost degrevlex.

The source states that this conjecture implies Fröberg's Hilbert-series conjecture, but gives no resolution status for the conjecture itself.