Aldous's spectral-gap conjecture for the symmetric simple exclusion process

Let $G=(V,E)$ be a finite graph with positive edge rates $r_{xy}$, and let $L$ be the generator of the symmetric simple exclusion process on the $n$-particle configuration space $\Omega_n$. Write $\lambda(n)>0$ for the smallest positive eigenvalue of $L$ on $l^2(\Omega_n)$. Aldous's spectral-gap conjecture.

$$\lambda(n)=\lambda(1),\qquad 1\leq n\leq\lvert V\rvert-1.$$

Thus the relaxation spectral gap of the exclusion process is independent of particle number and equals the random-walk gap. The supplied text presents this as a conjecture attributed to Aldous, based on discussions with Diaconis, and gives no resolution.