Noncyclicity conjecture for stationary-measure solutions under the backward shift

Let $\mu$ be a probability measure satisfying the Blaschke condition, with support generating a discrete subgroup of $PSU(1,1)$. Let $f_\nu(z)$ be a solution of the equation referred to in the source as main equation, simp. Noncyclicity conjecture. The function $f$ is $S^*$-noncyclic for every $p<1$. The conjecture is motivated by the classification of closed backward-shift-invariant subspaces of $H^p(\mathbb{D})$ for $0<p<1$; the supplied text gives no resolution status.