Polynomial mixing time of the triangle-solitaire Markov chain

Prove that the Markov chain on triangle bases induced by the triangle-solitaire dynamics has polynomial mixing time, thereby enabling uniform sampling of triangle bases and, via the established bijection, uniform sampling of 3-permutations avoiding the patterns 12 and 312.

Background

The triangle-solitaire dynamics defines a Markov chain on triangle bases. The paper cites that the diameter of the associated reconfiguration graph is O(n^3) and notes that the graph has strong connectivity properties.

The authors conjecture that the mixing time is polynomial. Establishing this would provide a method for uniformly sampling triangle bases; the bijection in the paper would then transfer such a sampling procedure to 3-permutations avoiding 12 and 312.