The projective-geometry embedding conjecture for clean tangled clutters

Let a clutter $\mathcal{C}$ embed $PG(k-2,2)$ when some subset of $\mathcal{C}$ is a duplication of the cuboid of $\operatorname{cocycle}(PG(k-2,2))$. A clutter is clean tangled if it is both clean and tangled.

Projective-geometry embedding conjecture. Every clean tangled clutter embeds a projective geometry over the two-element field.

This conjecture is connected with dyadic fractional packings of value two in clean tangled clutters. It is known when the setcore of the clutter has a simplicial convex hull, but remains open in general.