Rota–Heron–Welsh log-concavity conjecture for matroid characteristic coefficients

Let $\mathrm{M}$ be a matroid on a finite set $E$, and write $r+1$ for its rank. Let $\chi_{\mathrm{M}}(\lambda)$ be its characteristic polynomial, and let $w_k(\mathrm{M})$ be the absolute value of the coefficient of $\lambda^{r-k+1}$ in $\chi_{\mathrm{M}}(\lambda)$. Rota–Heron–Welsh conjecture. The sequence $w_k(\mathrm{M})$ is log-concave:

$$w_{k-1}(\mathrm{M})w_{k+1}(\mathrm{M})\leq w_k(\mathrm{M})^2\quad\text{for all }1\leq k\leq r.$$

In particular, the sequence is unimodal, so for some index $l$,

$$w_0(\mathrm{M})\leq w_1(\mathrm{M})\leq\cdots\leq w_l(\mathrm{M})\geq\cdots\geq w_r(\mathrm{M})\geq w_{r+1}(\mathrm{M}).$$

This conjecture extends log-concavity and unimodality questions for chromatic polynomials to arbitrary matroids. The source presents it as an open problem and notes that positivity of the coefficients is used to deduce unimodality from log-concavity.