Boundary behavior of the Böttcher map near the parameter a=-1

Prove that the Böttcher-map values at the critical point converge to i as the parameter a approaches -1 from any direction within the Klein combination locus.

Background

The paper proves that the upper and lower continuations of the canonical map have distinct limits at a=-1, while the auxiliary Böttcher maps evaluated at the critical point both converge to i when a approaches -1 within the unit disc. The authors conjecture that this convergence persists for arbitrary approaches to -1 lying inside the Klein combination locus, but state that the boundary of the locus near -1 is not sufficiently understood to establish it.

References

It is natural to conjecture that this statement holds more generally for $a$ approaching $-1$ from any direction within $$, but we do not yet know enough about the boundary of $K$ in a neighbourhood of $-1$ to prove this.

— Tessellating the discreteness locus for the modular mating family of correspondences  (2608.17243 - Bullett et al., 18 Aug 2026) in Section 2, immediately before Proposition 2.?? and following Lemma varphi-1