Insensitivity of Khovanov homology under rim surgery

We show that rim surgery on a smooth, oriented, properly embedded surface in $B^4$ does not change the map on Khovanov homology induced by the surface, answering a question raised by Hayden and Sundber. More generally, the same insensitivity holds for a class of local surgery operations that we call local annulus replacements. Combined with the work of Hayden--Sundberg, this result yields exotic pairs of surfaces in $B^4$ that cannot be related by any sequence of local annulus replacements. We give two proofs, both making essential use of Khovanov skein lasagna modules.