Aldous's spectral-gap conjecture for the interchange process

Let $X_N=\{0,1\}^N$ be the vertex set of the $N$-dimensional hypercube, let $X_{N,2}=\{\{i,j\}:1\leq i<j\leq N\}$ be the set of coordinate pairs, and let $q:X_{N,2}\to[0,\infty)$ be a rate function. Define the random-walk and interchange-process generators by

$$\Omega^{\text{RW}}_N(q)=\sum_{\{i,j\}\in X_{N,2}}q(\{i,j\})\Delta_{N,(i,j)},\qquad \Omega^{\text{IP}}_N(q)=\sum_{\{i,j\}\in X_{N,2}}q(\{i,j\})\widehat{\Delta}_{N,(i,j)}.$$

Write $\gamma^{\text{RW}}_N(q)=\gamma(\Omega^{\text{RW}}_N(q))$ and $\gamma^{\text{IP}}_N(q)=\gamma(\Omega^{\text{IP}}_N(q))$ for their spectral gaps. Aldous's conjecture. For every $N\geq2$ and every $q:X_{N,2}\to[0,\infty)$,

$$\gamma^{\text{IP}}_N(q)=\gamma^{\text{RW}}_N(q).$$

The conjecture asserts that the interchange process has the same spectral gap as the associated random walk for every nonnegative choice of transposition rates; the general resolution status is not specified in the source.