Mass-point conjecture for steady states of dual Markov chains

Let $\boldsymbol{\kappa}_\lambda$ be the nonnegative left eigenvector of the dual multiple stochastic matrices associated with a measure $\mu$ and parameter $\lambda$. A mass point is a point satisfying $\mu\big(\{\lambda\}\big)\neq 0$. Mass-point conjecture. The vector $\boldsymbol{\kappa}_\lambda$ belongs to $\ell_1$ if and only if $\lambda$ is a mass point of $\mu$. Hence, if $\mu$ is absolutely continuous and has no singular part, $\boldsymbol{\kappa}_\lambda\notin\ell_1$ and a recurrent Markov chain is null recurrent. The conjecture extends the corresponding birth-and-death-chain phenomenon to the multiple orthogonal setting and links finite steady states to mass points of the measure.