Backwards critical 3-relations on the boundary

Determine whether any backwards critical 3-relation parameters lie on the boundary of the Klein combination locus.

Background

Backwards critical relations are shown to lie outside the Klein combination locus, but numerical plots display sequences of such parameters near its boundary. The authors note that even if some of these parameters lie on the boundary, the associated correspondences may fail to be discrete because their critical cycles can force incompatible fundamental-domain boundaries.

References

It is an open question whether there are any backwards $3$-critical relations on $\partial$.

— Tessellating the discreteness locus for the modular mating family of correspondences  (2608.17243 - Bullett et al., 18 Aug 2026) in Remark 3.??, Section 3.1 “Critical relations on $\partial$”