Brändén–Huh's characterization of matroid bases by 8/7-Rayleigh polynomials

Let $[n]=\{1,\ldots,n\}$ and let $J$ be a non-empty subset of $\{0,1\}^{n}$. For $J$, write $f_J$ for its generating function, and call a polynomial homogeneous $c$-Rayleigh when it satisfies the $c$-Rayleigh inequalities for all positive real inputs.

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.

This conjecture proposes an analytic characterization of matroid basis sets. The source presents it as a conjecture and gives no resolution, so it remains open.