Lasagna s-invariants and nonvanishing for Stein 2-handlebodies

Prove that for any knot K and integer n < TB(K), the n-trace X_n(K) satisfies s(X_n(K);0) = 0 and s(X_n(K);1) = s(K) − n, and consequently the Khovanov skein lasagna module S^2_0(X_n(K)) is nonzero. More generally, prove that every Stein 2-handlebody X has s(X;0) > −∞ and nonvanishing Khovanov skein lasagna module.

Background

The authors paper the Khovanov–Rozansky gl_2 skein lasagna module and introduce lasagna s-invariants as smooth invariants of 4-manifolds. They obtain several nonvanishing and genus bound results but note limits of current techniques for 4-manifolds with b_2^+ > 0.

To guide further progress, they propose a conjecture asserting specific values of lasagna s-invariants for traces X_n(K) below the Thurston–Bennequin threshold and, in general, nonvanishing skein lasagna modules for any Stein 2-handlebody. Establishing this would yield broad nonvanishing results and sharpen shake genus bounds.