The extremal-eigenstate convergence conjecture for the symmetric open baker

Consider the symmetric quantized open baker with extremal nontrivial eigenvalues $\lambda_+$ and $\lambda_-$. Let $\mu^\Sigma_{\vP_+}$ and $\mu^\Sigma_{\vP_-}$ be the corresponding Bernoulli eigenmeasures of the symbolic dynamics, with the weights defined from the eigenvectors in the source. Let $(\psi_N)_{N\to\infty}$ be a sequence of eigenstates with eigenvalues $\lambda_N$.

Extremal-eigenstate convergence conjecture. If

$$|\lambda_N|\to|\lambda_+|,$$

then $(\psi_N)$ converges to the semiclassical measure $\mu^\Sigma_{\vP_+}$; respectively, if

$$|\lambda_N|\to|\lambda_-|,$$

then $(\psi_N)$ converges to the semiclassical measure $\mu^\Sigma_{\vP_-}$.

The conjecture is based on the observed small degeneracies of eigenvalues near the circles of radii $|\lambda_+|$ and $|\lambda_-|$, together with explicit factorization for the extremal tensor-product eigenstates. It remains an unproved proposal in the source.