Permutation Tutte inequality for bipartite graphs of minimum degree two

Prove that every bipartite graph H with minimum degree at least 2 satisfies the multiplicative permutation Tutte inequality \(\widetilde{T}_H(2,0)\widetilde{T}_H(0,2)\geq \widetilde{T}_H(1,1)^2\).

Background

The paper establishes several lower bounds for products of evaluations of the permutation Tutte polynomial and uses a transfer lemma to deduce corresponding inequalities for the Tutte polynomial of matroids. The conjectured inequality would extend the paper’s results from certain degree ranges to all bipartite graphs whose minimum degree is at least two. Since $\widetilde{T}_H(1,1)=1$, the conjecture is equivalent in this setting to proving $\widetilde{T}_H(2,0)\widetilde{T}_H(0,2)\geq 1$.