Merino–Welsh inequality for simple co-simple matroids

Prove that every simple co-simple matroid M satisfies the multiplicative Merino–Welsh inequality \(T_M(2,0)T_M(0,2)\ge T_M(1,1)^2\).

Background

A simple co-simple matroid has no loops or parallel elements and whose dual has no loops or parallel elements. The paper notes that Conjecture 7.2 would follow immediately from Conjecture 7.1 via the local basis exchange graph and the transfer lemma: simplicity and co-simplicity ensure that every such local basis exchange graph has minimum degree at least 2. Thus this conjecture extends the verified classes of matroids beyond the prescribed circuit-length conditions proved earlier in the paper.