Welsh–Mason log-concavity conjecture for independent-set numbers of matroids

Let $\mathrm{M}$ be a matroid on a finite set $E$, write $r+1$ for its rank, and call a subset of $E$ independent if no element belongs to the closure of the other elements. Let $f_k(\mathrm{M})$ be the number of independent subsets of $E$ with cardinality $k$. Welsh–Mason conjecture. The sequence $f_k(\mathrm{M})$ is log-concave:

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

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

$$f_0(\mathrm{M})\leq f_1(\mathrm{M})\leq\cdots\leq f_l(\mathrm{M})\geq\cdots\geq f_r(\mathrm{M})\geq f_{r+1}(\mathrm{M}).$$

This is the related conjecture attributed in the source to Welsh and Mason concerning the numbers of independent sets of each cardinality in a matroid. Its resolution status is not supplied in the provided text.