Inconsequential results on the Merino-Welsh conjecture for Tutte polynomials

The Merino-Welsh conjectures say that subject to conditions, there is an inequality among the Tutte-polynomial evaluations $T(M;2,0)$, $T(M;0,2)$, and $T(M;1,1)$. We present three results on a Merino-Welsh conjecture. These results are “inconsequential” in the sense that although they imply a version of the conjecture for many matroids, they seem to be dead ends.