Near-linear mixing-time conjecture for uniform quadrangulations

Let $Q_n$ be a uniform quadrangulation with $n$ vertices, let $Q_n^\dagger$ be its dual, and let $\tau_{Q_n}(\varepsilon)$, $\tau_{Q_n^\dagger}(\varepsilon)$ denote uniform mixing times; write $\tau^{\mathrm{TV}}$ for total-variation mixing times and $\tau^{\mathrm{rel}}$ for relaxation times. Near-linear mixing-time conjecture. For every $\delta,\varepsilon>0$, with probability tending to $1$ as $n\to\infty$,

$$\tau_{Q_n}(\varepsilon)\in[n^{1-\delta},n^{1+\delta}].$$

The same assertion holds for $\tau_{Q_n^\dagger}(\varepsilon)$, $\tau^{\mathrm{TV}}_{Q_n}(\varepsilon)$, $\tau^{\mathrm{TV}}_{Q_n^\dagger}(\varepsilon)$, $\tau^{\mathrm{rel}}_{Q_n}$, and $\tau^{\mathrm{rel}}_{Q_n^\dagger}$. This formalizes the simulation-based expectation that the relevant mixing and relaxation times are $n^{1+o(1)}$, substantially sharper than the available upper bounds.