The two-bases conjecture for the Tutte polynomial

Let $M$ be a matroid, and let $T_M(x,y)$ denote its Tutte polynomial. Say that $M$ contains two disjoint bases if it has bases $B_1$ and $B_2$ with $B_1\cap B_2=\varnothing$; alternatively, its ground set may be the union of two bases if there are bases $B_1,B_2$ with $E(M)=B_1\cup B_2$. Two-bases conjecture. If $M$ contains two disjoint bases or its ground set is the union of two bases, then

$$\max\{T_M(2,0),T_M(0,2)\}\geq T_M(1,1).$$

This is described as a weaker and potentially more tractable form of the Merino–Welsh conjecture. The paper proves the inequality for several classes, but leaves the stated general case as a conjecture.