Conjecture on Bernstein interpolation and operator-theoretic properties of geometrically regular weighted shifts

Let $p>1$ and let $(N,D)$ satisfy $-1<N<1$ and $-1<D<1$. Let $W_\alpha$ be the weighted shift with weight sequence $\alpha$ given by

$$\alpha_n=\sqrt{\frac{p^n+N}{p^n+D}},\qquad n=0,1,2,\ldots.$$

The sectors refer to the division of the $(N,D)$-square given in the paper's diagram.

The conjecture. In Sector II, except on its boundary with Sector I, no $\alpha$ is interpolated by a Bernstein function; in Sector III, $W_\alpha$ is subnormal, and except on its boundary with Sector II, no shift is MID; and, other than as specified above, there are no MID, subnormal, or completely hyperexpansive shifts in the square.

These conjectures seek to complete the classification of the geometrically regular weighted shifts by extending the properties established in the preceding theorem. The conjectured exclusions concern Bernstein interpolation, MID shifts, subnormality, and complete hyperexpansivity in the remaining regions; their resolution is not supplied here.