Khovanov skein lasagna modules with $1$-dimensional inputs

We construct a variant of Khovanov skein lasagna modules, which takes the Khovanov homology in connected sums of $S^1\times S^2$ defined by Rozansky and Willis as the input link homology. To carry out the construction, we prove functoriality of Rozansky-Willis's homology for cobordisms in a class of $4$-manifolds that we call $4$-dimensional relative $1$-handlebody complements, by using, as a bypass, an isomorphism proved by Sullivan--Zhang relating the Rozansky-Willis homology and the classical Khovanov skein lasagna module of links on the boundary of $D^2\times S^2$. Along the way, we also present new results on diffeomorphism groups, on Gluck twists for Khovanov skein lasagna modules, and on the functoriality of $\mathfrak{gl}_2$ foams.