Distinct equivariant intersection forms for immersed Z-surfaces

Determine whether there exist pairs of immersed Z-surfaces in a simply-connected 4-manifold that have the same genus and the same numbers of positive and negative transverse double points but whose exteriors have non-isometric equivariant intersection forms. In particular, determine whether every immersed Z-surface in S^4, or in D^4 with boundary an Alexander polynomial one knot, necessarily has exterior equivariant intersection form isometric to λ_{c_+,c_-} ⊕ 𝓗_2^{⊕ g}, where λ_{c_+,c_-}=((t−1)(t^{-1}−1))^{⊕ c_+} ⊕ (−(t−1)(t^{-1}−1))^{⊕ c_-} and 𝓗_2 is the standard 2×2 hyperbolic form over Z[t^{±1}].

Background

The paper classifies immersed Z-surfaces in simply-connected 4-manifolds using the equivariant intersection form of the exterior together with secondary data. For embedded Z-surfaces, the equivariant intersection form is known to determine the surface up to equivalence in many cases, and there are strong constraints on what forms can arise.

In the immersed setting, the authors obtain comprehensive classification and uniqueness results in several special cases (e.g., genus zero with a single double point, and surfaces in D^4 or S^4), but they note that it is unclear if more exotic equivariant intersection forms can occur when the number of double points increases.

Resolving this question would clarify whether the standard form λ{c+,c_-} ⊕ 𝓗_2^{⊕ g} is universal in S^4 and for Alexander polynomial one boundaries in D^4, and more generally, whether the equivariant form imposes constraints on possible immersed Z-surfaces beyond the cases treated in the paper.