The Merino–Welsh conjecture for bipartite graphs of minimum degree at least two

Let $H$ be a bipartite graph, and let $\widetilde{T}_H(x,y)$ denote its normalized Tutte polynomial. Assume that every vertex of $H$ has degree at least $2$. Merino–Welsh conjecture. Then

$$\widetilde{T}_H(2,0)\widetilde{T}_H(0,2)\geq \widetilde{T}_H(1,1)^2.$$

This is a bipartite-graph formulation of the Merino–Welsh inequality. The paper proves the inequality for the local basis exchange graphs arising from matroids under stronger degree conditions, while the minimum-degree-two case stated here remains open.