Recurrence classification conjecture for dual multiple-orthogonal-polynomial Markov chains

Let $\boldsymbol{\kappa}_\lambda$ be the left eigenvector associated with the dual Markov chains, and let $\lambda$ be the parameter used to construct them. The chains are called transient when their expected return behavior is nonrecurrent, positive recurrent when they admit a normalizable steady state, and null recurrent when they are recurrent but have infinite expected return times. Recurrence classification conjecture. If

$$\int_\Delta \frac{w_1(x)}{1-\frac{x}{\lambda}}\operatorname{d}\mu(x)$$

converges, both dual Markov chains are transient and $\lambda$ is not a mass point. If the integral diverges, then both chains are ergodic when $\lambda$ is a mass point and are null recurrent when $\lambda$ is not a mass point. Thus, in the former recurrent case the expected return times are finite, whereas in the latter they are infinite. This conjecture refines the recurrence criterion by relating transience, ergodicity, and null recurrence to mass points of the measure.