Nonnegativity conjecture for the omega invariant of a matroid

Let $M$ be a matroid of rank $r$, and write $g_M(t)=\sum_{i\geq 1}g_i(M)t^i$ when the ground set is nonempty. Define the omega invariant by $\omega(M)=g_r(M)$, the coefficient of $t^r$ in $g_M(t)$; in particular, $\omega(M)=0$ if $r>n-r$. Omega nonnegativity conjecture. For any matroid $M$, we have

$$\omega(M)\geq 0.$$

This is the specific case of the nonnegativity conjecture for the coefficients of $g_M(t)$ concerning the omega invariant, and its resolution is not established in the supplied text.