Aldous-Diaconis spectral-gap conjecture for the symmetric simple exclusion process

Let $\Lambda$ be a finite graph, and let $L$ be the generator of the symmetric simple exclusion process with positive jump rates $J_{xy}$ on its edges. Let $H_\Lambda^{(n)}$ be the invariant subspace of configurations with exactly $n$ particles, and let $\lambda(n)$ denote the smallest positive eigenvalue of $L\vert_{H_\Lambda^{(n)}}$. Aldous-Diaconis conjecture.

$$\lambda(n)=\lambda(1),\quad 1\leq n\leq\lvert\Lambda\rvert-1.$$

Thus the spectral gap of the exclusion process is independent of the particle number, matching the one-particle spectral gap. The source attributes the conjecture to Aldous, while noting that Aldous said it arose in a conversation with Diaconis; the supplied text gives no resolution status.