Pointwise stabilization equivalence for discrete-time stationary matrix-valued Markovian chains

Let $(\boldsymbol{S},\Xi)$ be a discrete-time stationary matrix-valued Markovian chain, with consistent convergence and consistent exponential convergence understood in the sense of the paper. Pointwise stabilization conjecture. Without assuming $\hat{\rho}(\boldsymbol{S})=1$, $(\boldsymbol{S},\Xi)$ is consistently convergent if and only if it is consistently exponentially convergent.

The result is proved in the paper under the additional condition $\hat{\rho}(\boldsymbol{S})=1$; the conjecture asks whether that condition can be removed.