Linear quotients imply ridge-chordality for clutters

Let ${\mathcal C}$ be a $d$-clutter, and let $I_{\overline{\mathcal C}}$ be the circuit ideal of its complement. A ridge is a $(d-1)$-element set contained in a circuit, and it is simplicial if the induced subclutter on the ridge together with all vertices completing it to a circuit is complete. The clutter ${\mathcal C}$ is ridge-chordal if it admits a sequence of simplicial ridges whose successive removals delete all circuits.

Linear-quotients ridge-chordality conjecture. If $I_{\overline{\mathcal C}}$ has linear quotients, then ${\mathcal C}$ is ridge-chordal.

The source explicitly presents this implication as an open question. Ridge-chordality is known to imply a linear resolution over every field, but the converse from linear quotients is not established there.