Brändén–D’León relaxation conjecture for the weak half-plane property

A matroid has the weak half-plane property when its set of bases agrees with the support of a stable polynomial. A relaxation of a matroid is obtained by relaxing a circuit-hyperplane. Brändén–D’León's conjecture. If a matroid $M$ has the weak half-plane property, then every relaxation of $M$ also has the weak half-plane property. The paper gives a relaxation of the matroid $P_8$ that does not have the weak half-plane property, disproving the conjecture.