Conjecture on affiliation of the incoming wave operator for the radial part of $SL(2,\mathbb{R})$

Let $H_{\mu,\nu}$ be the operator parametrized by $\mu,\nu\geq0$, let $H_{\rm D}$ be the Dirichlet reference operator, and let $W_-(H_{\mu,\nu},H_{\rm D})$ denote the incoming wave operator. Let $\mathscr E$ be the $C^*$-algebra used in the scattering framework. Affiliation conjecture. For any $\mu,\nu\geq0$, one has

$$W_-(H_{\mu,\nu},H_{\rm D})\in\mathscr E.$$

The conjecture would establish the required algebraic affiliation of the incoming wave operator and support the associated topological index-theoretic formulation of Levinson's theorem. The source states that this affiliation has not been proved because of the complicated structure of the ${}_2F_1$-function, and notes that the final index theorem does not depend on the conjecture.