The strengthened polynomial log-concavity conjecture for independent-set polynomials

Let $M=(E,\mathcal{I})$ be a matroid with rank $r_M$, and let $f_k(M)$ be its degree-$k$ independent-set polynomial

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

For $0<k<r_M$, the strengthened polynomial log-concavity conjecture.

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

This is motivated by Dowling's polynomial conjecture and the second inequality in Mason's conjecture, and is attributed in the source to an implicit conjecture of Zhao. The supplied text does not state a resolution, so its status remains open.