Merino–Welsh inequality relating spanning trees and orientations

Prove for every finite graph G that T_G(1,1) ≤ max{T_G(2,0), T_G(0,2)}, which asserts that the number of spanning trees is at most the maximum of the number of acyclic orientations and the number of totally cyclic orientations.

Background

Merino and Welsh formulated a conjecture bounding the number of spanning trees of a graph by the maximum of the counts of acyclic and totally cyclic orientations, expressible via evaluations of the Tutte polynomial.

The memoir explicitly notes that this inequality remains open since its proposal in 1999, highlighting its status as a longstanding conjecture in Tutte polynomial theory.