Depth criterion for squarefree lexsegment ideals

Let $S$ be a polynomial ring, let $u=x_1x_{i_2}\cdots x_{i_q}$ and $v=x_{j_1}\cdots x_{j_q}$ with $j_1\geq 2$, and let $I=(L(u,v))$ be the squarefree lexsegment ideal determined by $u$ and $v$. Assume the indices are ordered as in the displayed squarefree monomials.

Depth criterion conjecture. One has

$$\depth(S/I)=q-1$$

if and only if

$$x_{i_2-1}x_{i_3-1}\cdots x_{i_{q-1}}x_n\geq v.$$

This is proposed as a general criterion based on results for squarefree lexsegment ideals generated in small degrees. The source presents it among open questions and conjectures, so its general validity remains open.