Stein 2-handlebody conjecture for Khovanov skein lasagna invariants

Let $K$ be a knot, let $TB(K)$ denote its Thurston–Bennequin number, and let $X=X_n(K)$ be the $n$-trace on $K$ with $n<TB(K)$. Let $s(X;0)$ and $s(X;1)$ be the corresponding lasagna $s$-invariants, and let $s(K)$ be the knot invariant. Stein manifold conjecture. If $n<TB(K)$, then

$$s(X;0)=0$$

and

$$s(X;1)=s(K)-n.$$

In particular, $\mathcal S_0^2(X)\ne0$. More generally, every Stein $2$-handlebody $X$ has $s(X;0)>-\infty$ and nonvanishing Khovanov skein lasagna module. The claim would extend the presently observed sensitivity of these invariants to Stein manifolds and provide nonvanishing examples beyond the cases currently computable; its general validity remains open.