Tutte-polynomial inequality for simple co-simple matroids

Let $M$ be a matroid. A matroid is simple if it has no loops or parallel elements, and co-simple if its dual has no loops or parallel elements. Simple co-simple matroid conjecture. Every simple co-simple matroid satisfies

$$T_M(2,0)T_M(0,2)\geq T_M(1,1)^2.$$

This would follow immediately from the preceding bipartite graph conjecture via the transfer argument. It is an open matroidal strengthening of Jackson's inequality, with the supplied text giving no resolution.