Equality of multiplier algebras for single-crossing embeddings f_{r,−r}

Determine whether, for all r, s ∈ (0,1), the multiplier algebras M_{f_{r,−r}} and M_{f_{s,−s}} are equal, where f_{r,−r} denotes the two-dimensional analytic disc embedding with a single boundary self-crossing at ±1 constructed using b_r(z) = (z − r)/(1 − r z).

Background

The thesis analyzes a family of attached analytic discs parameterized by r, featuring exactly one boundary self-crossing, and shows rigidity constraints on possible isomorphisms of their multiplier algebras.

A natural question is whether these multiplier algebras are in fact all identical across the family, which would strengthen the rigidity results or, if false, provide counterexamples to isomorphism under identical crossing type.