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 $\mathrm{KhR}_2$ 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 $\mathrm{KhR}_2$ 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 $\mathrm{KhR}_2$ skein lasagna module of $S^2 \times S^2$ is trivial, confirming a conjecture of Manolescu.