Existence of b2^+>0 4-manifolds with finite lasagna s-invariants or nonvanishing skein lasagna modules

Construct a simply-connected closed smooth 4-manifold X with b_2^+(X) > 0 such that either the lasagna s-invariants s(X;·) are finite or the Khovanov skein lasagna module S^2_0(X) is nonzero.

Background

The authors’ methods currently do not yield examples of closed 4-manifolds with b_2^+>0 having finite lasagna s-invariants or nonvanishing skein lasagna modules. Producing such an example would have strong consequences: X#CP^2 would be an exotic closed 4-manifold, giving an analysis-free path to detecting exotic smooth structures in the closed case.