Wagner's conjecture on S- and I-Rayleigh regular matroids

Let $M$ be a regular matroid. The properties S-Rayleigh and I-Rayleigh are the corresponding negative-dependence properties for matroids.

Wagner's conjecture. Every regular matroid is S- and I-Rayleigh.

Regular matroids include graphic and cographic matroids. A positive answer would confirm the conjectures of Kahn and of Grimmett and Winkler asserting that the unrooted spanning-forest distribution on a graph is negatively correlated; the conjecture's resolution is not specified in the source.