Exact spectral-gap relation for the multi-urn backgammon model

Let $P^{(m)}$ be the transition matrix of the $m$-urn backgammon model and let $g(P^{(m)})$ denote its spectral gap. The model has $n$ balls distributed among $m$ urns, with the notation and dynamics defined above.

Exact spectral-gap conjecture. The inequality relating the $m$-urn and two-urn gaps is an equality:

$$g(P^{(m)})=\frac{1}{m-1}g(P^{(2)}).$$

Numerical diagonalization for the tested ranges of $m$ and $n$ supports the equality; the preceding argument establishes only the corresponding inequality, so the general exact relation remains unproved in the source.