Non-idealness conjecture for sufficiently wise-intersecting clutters

A clutter is a family of sets none of which contains another. It is $k$-wise intersecting if every subset of at most $k$ members has a common element, yet no element belongs to all members. It is ideal if its primal covering linear program has an integral optimum for every integral nonnegative cost vector.

Non-idealness conjecture. There exists an integer $k\geq 4$ such that every $k$-wise intersecting clutter is non-ideal.

Intersecting clutters can nevertheless be ideal, so the conjecture predicts that sufficiently strong finite-wise intersection forces non-idealness. The source further conjectures that $k=4$ suffices; the claim is presented as open.