Cyclic Markov-chain GUE conjecture for random-word shapes

Let $M$ be an indecomposable, doubly stochastic matrix indexed by an alphabet of size $k$, satisfying

$$M_{a,b}=M_{a+1,b+1},$$

where $k+1\equiv1$. A word of length $N$ is generated by the Markov chain with transition matrix $M$, and its shape is the partition obtained from the word by the relevant subsequence-shape correspondence. Cyclic Markov-chain GUE conjecture. As $N\to\infty$, the distribution of the shape of the generated word converges locally to the distribution of the spectrum of a traceless $k\times k$ Gaussian unitary ensemble matrix. The conjecture formalizes the numerical observation that cyclic symmetry of the transition matrix produces the same limiting fluctuations as the uniform model, while asymmetric Markov chains can have a different limit.