Almost all matroids have no monotonicity violation in matroid bingo

Prove that the asymptotic proportion of matroids on n-element ground sets for which matroid-bingo monotonicity holds—namely, for all circuits C1 and C2 with |C1| > |C2|, the winning probabilities satisfy β_{C1} < β_{C2}—tends to 1 as n goes to infinity.

Background

In the matroid bingo setting, each circuit C of a matroid M corresponds to a bingo card, and β_C denotes its probability of winning. The monotonicity property requires that larger circuits have strictly smaller winning probabilities than smaller circuits, preventing any monotonicity violations.

Since paving matroids have no monotonicity violations and are conjectured to comprise almost all matroids asymptotically, the authors formulate the conjecture that almost all matroids satisfy monotonicity.