The shifting conjecture for non-reverse-lexicographic term orders

Let $S$ be a polynomial ring, let $\succ$ be a term order on $S$ other than the graded reverse lexicographic order, and let $k\geq 2$ be the smallest degree on which $\succ$ and the reverse lexicographic order disagree. For each finite index $i$, let $m_i$ be the $i$th squarefree monomial of degree $k$ with respect to the reverse lexicographic order, and set

$$I_i:=\langle m_1,\ldots,m_i\rangle.$$

Let $i_0\geq 1$ be the smallest index for which $I_{i_0}$ is not a US$\succ$I. The shifting conjecture. Then

$$\Delta_{\succ}(I_{i_0})\neq I_{i_0}.$$

This conjecture would give a negative answer to whether a term order other than reverse lexicographic order can satisfy $\Delta_{\succ}(I)=I$ for every squarefree strongly stable ideal. The preceding results establish the corresponding affirmative statement for US$\succ$I ideals and show that USLIs are fixed by every term order, while the proposed assertion remains unproved in the source.