Merino–Welsh conjecture for graphs

Let $G$ be a connected graph without loops or bridges. Write $\tau(G)$ for the number of spanning trees of $G$, $b1(G)$ for the number of acyclic orientations of $G$, and $b1^*(G)$ for the number of totally cyclic orientations of $G$.

Merino–Welsh conjecture. The inequality

$$\max(\alpha(G),\alpha^*(G))\geq\tau(G)$$

should hold.

The three quantities are evaluations of the Tutte polynomial: $\tau(G)=T_G(1,1)$, $\alpha(G)=T_G(2,0)$, and $\alpha^*(G)=T_G(0,2)$. The conjecture is known for several graph classes, including sparse or dense graphs, but is not resolved for all connected graphs without loops and bridges.