Fast mixing conjecture for face flips on the triangle lattice

Let $T_{m,n}$ be a parallelogram-shaped triangle-lattice crease pattern with $m$ and $n$ triangles along its two adjacent sides, and consider the face-flip Markov chain on $T_{m,n}$.

Fast mixing conjecture. The mixing time of the face-flip Markov chain on $T_{m,n}$ is $O(mn \log(mn))$.

The conjecture is motivated by repeated simulations on relatively large examples. The face-flip graph is known to be connected and to have diameter $O(mn)$, but proofs of this optimal-order mixing-time bound remain elusive.