Existence of every rational-slope wall for scattering diagrams with negative coefficients

Determine whether, for rank-2 generalized cluster scattering diagrams with initial wall-function coefficients that are allowed to be negative, there are walls of every rational slope within the Badlands, as asserted by Corollary B.2 when the initial coefficients are nonnegative.

Background

Appendix B proves that, when the coefficients of the two initial wall-functions are nonnegative, every rational slope within the Badlands supports a wall. The proof uses tight gradings whose weights are then positive, ensuring that the corresponding wall-function coefficients do not vanish.

The authors note that allowing negative initial coefficients can cause cancellation among tight-grading weights, so wall-function coefficients may vanish. They explicitly leave unresolved whether this cancellation can make the conclusion of Corollary B.2 fail; equivalently, it is unknown whether every rational slope must still occur as a wall in the Badlands under signed coefficients.