  We exhibit infinite families of annular links for which the maximum non-zero
annular Khovanov grading grows infinitely large but the maximum non-zero
annular Floer-theoretic gradings are bounded. We also show this phenomenon
exists at the decategorified level for some of the infinite families. Our
computations provide further evidence for the wrapping conjecture of
Hoste-Przytycki and its categorified analogue. Additionally, we show that
certain satellite operations cannot be used to construct counterexamples to the
categorified wrapping conjecture. We also extend the Batson-Seed link splitting
spectral sequence to the setting of annular Khovanov homology.
