Merino–Welsh conjectures for matroids

Let $M$ be a matroid without loops and coloops, and let $T_M(x,y)$ denote its Tutte polynomial.

Merino–Welsh conjectures. The following three inequalities should hold:

$$\max\left(T_M(2,0),T_M(0,2)\right)\geq T_M(1,1),$$ $$T_M(2,0)+T_M(0,2)\geq 2T_M(1,1),$$ $$T_M(2,0)T_M(0,2)\geq T_M(1,1)^2.$$

These are respectively the maximum, additive, and multiplicative versions of the graph conjecture. The multiplicative inequality implies the additive one, which implies the maximum inequality. The conjectures are known for several classes, including lattice path matroids and paving matroids for the additive version, while the general matroidal claims remain open.