Jahan–Zheng's conjecture on componentwise linear quotients

Let $S=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$, and let $I\subset S$ be a monomial ideal. For each $j\geq 0$, let $I_{\langle j\rangle}$ be the monomial ideal generated by the monomials of degree $j$ belonging to $I$. The ideal $I$ has componentwise linear quotients when every $I_{\langle j\rangle}$ has linear quotients; it has linear quotients when its minimal monomial generators can be ordered $u_1,\dots,u_m$ so that $(u_1,\dots,u_{j-1}):u_j$ is generated by variables for every $j=2,\dots,m$. Jahan–Zheng's conjecture. If $I$ has componentwise linear quotients, then $I$ has linear quotients. This converse to the known implication from linear quotients to componentwise linear quotients was described as widely open, with partial results available.