Projective-geometry embedding conjecture for ideal tangled clutters

A tangled clutter is an ideal clutter with covering number at least two such that every proper deletion minor has covering number one. A clutter $\mathcal{C}$ embeds $PG(r,2)$ if some subset of its members is a duplication of the cuboid associated with the cocycle space of the projective geometry $PG(r,2)$.

Projective-geometry embedding conjecture. There exists an integer $\ell\geq3$ such that every ideal tangled clutter embeds one of $PG(0,2),\ldots,PG(\ell-1,2)$.

The conjecture extends the known $Q_6$ property for a subclass of ideal tangled clutters, namely ideal minimally non-packing clutters with covering number two. It predicts that all ideal tangled clutters contain one of finitely many projective-geometric configurations.