Is torus symmetry forced? (O5)

Alpöge's map, Gallagher's family, the factory maps and Harish's map are all graded by a torus with weights of mixed sign. Is that forced?

The evidence is local. At Alpöge's map and at Gallagher's $F_4$, the Fong–Fable ledger reports the first-order Keller deformations of degree at most 7. Every one is either a composition move or an equivariant deformation. At $F$ a single residual direction is obstructed at second order, and none remains at $F_4$. The ledger calls this local, capped and two points deep, not a classification.^[1]

Why it matters: in the plane, Shaska's theorem makes every torus-symmetric Keller map an automorphism. If O5 holds in dimension two, there is no plane counterexample, so a positive answer to O5 would prove JC₂, through Shaska's plane theorem (itself unrefereed). A negative answer in dimension three would be a counterexample by a genuinely new mechanism.

1. ↑ D. Fong and Claude Fable 5, theorems register (theorem 8) and claims ledger, jacobianconjectures.com.