On minimum-size matroid counterexamples to the multiplicative Merino--Welsh inequality

Let $M$ be a matroid and let ${\text{Tutte}}_M(x,y)$ be its Tutte polynomial. The multiplicative Merino–Welsh inequality $${\text{Tutte}}_M(2,0){\text{Tutte}}_M(0,2)\ge {\text{Tutte}}_M(1,1{)}^2$$ was proved to be false for matroids by Beke, Csáji, Csikvári, and Pituk (\emph{Adv. Math.} $\mathbf{\text{446}}$ (2024), 109674). Let $U_{n,r}$ denote the rank-$r$ uniform matroid on $n$ elements, where $0<r<n$, and let $U_{n,r}^{(q)}$ denote the matroid obtained by replacing each element of $U_{n,r}$ with $q$ parallel copies. They constructed an infinite family of counterexamples of the form $U_{n,r}^{(2)}$, including $U_{33,22}^{(2)}$. In this paper, we prove that every $U_{n,r}^{(q)}$ with $q\ge 3$ satisfies the inequality. For $q=2$, we prove that the minimum number of elements in such a counterexample is $62$, attained by $U_{31,21}^{(2)}$. For loopless and bridgeless graphs, Noble and Royle (\emph{Eur. J. Comb.} $\mathbf{\text{38}}$ (2014), 24–35) showed that every edge of a minimum-size counterexample to the multiplicative Merino–Welsh conjecture lies in a series or parallel class of size two or three. For matroids, we further prove that every element of a minimum-size loopless and coloopless counterexample lies in exactly one series or parallel class of size two.