Spectral transition conjecture for the family of analytic Markov maps

Let $\mathcal{P}$ be the operator associated with the one-parameter family of analytic Markov maps, acting on the Hilbert space $\mathcal{H}$, and let $r\in[0,1]$ denote the parameter. The spectrum is denoted by $\operatorname{sp}(\mathcal{P})$. Spectral transition conjecture. For all $0\leq r<1$, the spectrum of $\mathcal{P}$ acting upon $\mathcal{H}$ is a countable subset of $[0,1]$ densely filling $[0,1]$ as $r\nearrow 1$. When $r=1$, the spectrum is purely continuous, meaning that there are no eigenvalues. The endpoint results preceding this statement establish a discrete spectrum at $r=0$ and a continuous component together with possibly finitely multiplicity eigenvalues at $r=1$; the conjecture asserts the transition behavior for $r<1$ and excludes eigenvalues at the endpoint $r=1$.