Conjecture on bichromatic triangles or quadrangles around a blue polygon

Consider a bicolored pseudoline arrangement, with pseudolines colored blue or red. Let $P$ be a blue $k$-gon, meaning a polygonal face or region bounded by blue pseudolines, and suppose that at least $k-4$ red pseudolines pass through $P$. A bichromatic triangle or quadrangle on the inside boundary is a triangular or quadrilateral face incident with the inside boundary of $P$ and supported by pseudolines of both colors.

Blue-polygon bichromatic-face conjecture. Given a blue $k$-gon $P$ with at least $k-4$ red pseudolines passing through, there exists a bichromatic triangle or quadrangle on its inside boundary.

The authors introduce this as an intermediate statement toward proving that every non-trivially bicolored arrangement contains a bichromatic triangle or quadrangle. They report that they could not prove it; its general status is open.