  A proof is given for the "only if" part of the result stated in the previous
paper 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 the
Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$, under
the 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 point
theorem in the proof of the main lemma and give an improved version of an
earlier result concerning the divisibility of $\mathrm {BMO}$-regularity.
