Kirby belts, categorified projectors, and the skein lasagna module of $S^2\times S^2$

We interpret Manolescu-Neithalath’s cabled Khovanov homology formula for computing Morrison-Walker-Wedrich’s ${\mathrm{K}\mathrm{h}\mathrm{R}}_2$ skein lasagna module as a homotopy colimit (mapping telescope) in a completion of the category of complexes over Bar-Natan’s cobordism category. Using categorified projectors, we compute the ${\mathrm{K}\mathrm{h}\mathrm{R}}_2$ skein lasagna modules of (manifold, boundary link) pairs $(S^2\times B^2,\tilde{\beta })$, where $\tilde{\beta }$ is a geometrically essential boundary link, identifying a relationship between the lasagna module and the Rozansky projector appearing in the Rozansky-Willis invariant for nullhomologous links in $S^2\times S^1$. As an application, we show that the ${\mathrm{K}\mathrm{h}\mathrm{R}}_2$ skein lasagna module of $S^2\times S^2$ is trivial, confirming a conjecture of Manolescu.