Moreno-Socias's generic weakly-grevlex ideal conjecture

Let $k$ be an infinite field, let $A$ be the polynomial ring under consideration, and let $A_{d_i}$ denote its degree-$d_i$ homogeneous component. For $s\in\mathbb{N}$ and $d_1,\dots,d_s\in\mathbb{N}$, consider tuples $(f_1,\dots,f_s)\in A_{d_1}\times\dots\times A_{d_s}$ and the ideal $I=(f_1,\dots,f_s)$. An ideal is weakly-grevlex if it is weakly-$w$-ideal for the graded reverse lexicographic monomial order $w$.

Moreno-Socias's conjecture. There is a non-empty Zariski-open subset $U\subseteq A_{d_1}\times\dots\times A_{d_s}$ such that every $(f_1,\dots,f_s)\in U$ generates an ideal $I=(f_1,\dots,f_s)$ that is weakly-grevlex.

The conjecture asserts that weakly-grevlex behavior is generic for ideals generated by homogeneous polynomials of prescribed degrees over an infinite field. If true, this would provide a broad class of inputs for which the Matrix-F5 algorithm can avoid the ambiguity caused by non-pivot columns and certify an approximate Gröbner basis.