Brändén and Huh's matroid characterization by homogeneous $\frac{8}{7}$-Rayleigh polynomials

Let $J$ be a non-empty subset of $\{0,1\}^{n}$, and let $f_J$ denote its generating function. A homogeneous polynomial is called $\frac{8}{7}$-Rayleigh when it has the corresponding $\frac{8}{7}$-Rayleigh property.

Brändén–Huh's conjecture. The following conditions are equivalent:

1. $J$ is the set of bases of a matroid on $[n]$.
2. The generating function $f_J$ is a homogeneous $\frac{8}{7}$-Rayleigh polynomial.

The conjecture seeks an intrinsic characterization of matroid basis sets through the Rayleigh property of their generating functions. The source attributes it to Brändén and Huh; no resolution is given here.