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A proof is given for the "only if" part of the result stated in the previouspaper of the series that a suitably nondegenerate Calder\'on-Zygmund operator$T$ is bounded in a Banach lattice $X$ on $\mathbb R^n$ if and only if theHardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$, underthe assumption that $X$ has the Fatou property and $X$ is $p$-convex and$q$-concave with some $1 < p, q < \infty$. We also get rid of a fixed pointtheorem in the proof of the main lemma and give an improved version of anearlier result concerning the divisibility of $\mathrm {BMO}$-regularity.

$\mathbf A_1$-regularity and boundedness of Calder\'on-Zygmund operators. II