  The rotation periods of Ap stars span more than five orders of magnitude. The
physical origin of this differentiation remains poorly understood. The
consideration of the most slowly rotating Ap stars represents a promising
approach to gain insight into the processes at play. The identification of
super-slowly rotating Ap (ssrAp) star candidates (defined as Ap stars that have
rotation periods longer than 50d) through systematic exploitation of the
available TESS observations of Ap stars is an effective approach to build a
sample devoid of magnetic bias. In our previous analyses of TESS Cycle 1 and
Cycle 2 data, we interpreted the Ap stars showing no photometric variability
over the 27-d duration of a TESS sector as being ssrAp star candidates. Here,
we apply the same approach to TESS Cycle 3 and Cycle 4 observations of Ap
stars. However, two issues may lead to spurious identification of ssrAp star
candidates. (1) A considerable fraction of the Ap stars in the existing lists
have erroneous or dubious spectral classifications. (2) The TESS data
processing may remove part of the variability signal. After critical evaluation
of these effects, we report the identification of 25 new ssrAp star candidates
and of 8 stars with moderately long periods. Combining this list with the lists
of ssrAp stars from Cycles 1 and 2 and with the list of ssrAp stars that were
previously known but whose lack of variability was not detected in our study,
we confirmed at a higher significance level the conclusions drawn in our
earlier work. These include the lower rate of occurrence of super-slow rotation
among weakly magnetic Ap stars than among strongly magnetic ones, the probable
existence of a gap between ~2 and ~3kG in the distribution of the magnetic
field strengths of the ssrAp stars, and the much higher rate of occurrence of
rapid oscillations in ssrAp stars than in the whole population of Ap stars.
