Global reversibility classification conjecture for involutive walks

Let $P^\lambda$ be an $n\times n$ stochastic matrix of the form $P^\lambda$. Let $J(n)$ denote the $n\times n$ matrix whose entries are all equal to $1/n$, and let the matrices in Theorem~ be the classified families of globally reversible involutive walks. Global reversibility conjecture. $P^\lambda$ is reversible if and only if it is one of the matrices in that theorem, or $P^\lambda=J(n)$. This conjecture seeks to classify all stochastic matrices of the form $P^\lambda$ that are reversible; the authors note that stochastic examples satisfying the detailed balance equations are known only with the stronger global reversibility property.