Hurwitz-stable flip--repair mixing conjecture

Let $\mu$ be a Hurwitz-stable distribution supported on one parity class. Consider the two-step parity flip--repair graph and its walk on each connected component: the walk flips a uniformly random coordinate to reach the opposite parity layer, then samples from $\mu$ conditioned on being one flip away from that intermediate point. Hurwitz-stable flip--repair mixing conjecture. On every connected component, this parity flip--repair walk has spectral gap $\Omega(1/n)$. If $\mu$ is flat, meaning that all positive atoms of $\mu$ have equal mass, then the same component walk satisfies a modified log-Sobolev inequality, or an ordinary log-Sobolev inequality, with constant $n\operatorname{polylog} n$. The conjecture concerns extending the spectral and functional-inequality behavior proved for skew-determinantal measures to general Hurwitz-stable distributions; disconnected supports are treated componentwise. Its status is open.