Orientation of nonideal clutters

Let $\mathcal{C}$ be a clutter on ground set V, and let $\mathcal{B}$ be its blocker. Is it true that $\mathcal{C}$ is nonideal if and only if there exist $p:{\mathcal C} \to V$ and $q: {\mathcal B} \to V$ such that

• $p(X)\in X$ for every $X \in {\mathcal C}$,
• $p(Y)\in Y$ for every $Y \in {\mathcal B}$,
• if $p(X)=q(Y)$, then $|X \cap Y|\gt1$?

Remarks

This was conjectured by T. Király. It can be shown by a polyhedral version of Sperner's Lemma [1] that no such functions exist for ideal clutters. It is also easy to see that it is enough to verify the conjecture for minimally non-ideal clutters. Pap[1] showed that the conjecture is true if the core of ${\mathcal C}$ is cyclic (see Lehman's theorem).

References

1. J. Pap, Integrality, complexity and colourings in polyhedral combinatorics, Ph.D. thesis, pdf link