Polynomial mixing-time conjecture for the flip walk on sphere triangulations

Let $(T_n(k))_{k \geq 0}$ denote the flip walk on triangulations of the sphere with $n$ vertices, and let its mixing time be measured with respect to the uniform distribution on such triangulations. Polynomial mixing-time conjecture. The mixing time of $(T_n(k))_{k \geq 0}$ is polynomial in $n$. The paper establishes a lower bound on the mixing time but does not prove a polynomial upper bound; obtaining such an upper bound is posed as an open question.