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.