Dowling's polynomial conjecture for independent-set polynomials of matroids

Let $M=(E,\mathcal{I})$ be a matroid with $|E|=n$ and rank $r_M$. For $0\leq k\leq r_M$, define the homogeneous multivariate independent-set polynomial

$$f_k(M)=\sum_{I\in\mathcal{I},\,|I|=k}\left(\prod_{x_i\in I}x_i\right).$$

For polynomials in $\mathbb{R}[x_1,x_2,\ldots,x_n]$, write $f\geq g$ when $f-g$ has nonnegative coefficients. Dowling's polynomial conjecture. For every $0<k<r_M$,

$$f_k^2(M)\geq f_{k-1}(M)f_{k+1}(M).$$

The conjecture strengthens Mason's ordinary log-concavity conjecture and implies its first inequality. Dowling proved it for $k\leq 7$; the paper claims a complete solution using Lorentzian polynomials.