One of the most challenging aspects of the Ap stars is the
extreme differentiation of their rotation periods, which 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 responsible for the rotational
braking to which the Ap stars are subject. However, historically,
the study of these stars focused primarily on the most strongly
magnetic among them. This bias introduced an ambiguity in the
conclusions that could be drawn, as it did not allow the distinction
between the rotational and magnetic effects, nor the investigation
of possible correlations between rotational and magnetic
properties. We previously showed that the identification
of super-slowly rotating Ap (ssrAp) star candidates (defined as Ap
stars that have rotation periods $ through systematic exploitation of the available TESS photometric observations of Ap stars is an effective approach to build a sample devoid of magnetic bias. This approach rests on the presence of brightness spots on the surface of Ap stars that are not distributed symmetrically about their rotation axes and show long-term stability, hence are responsible for photometric variations with the stellar rotation period. In our previous analyses of TESS Cycle 1 and Cycle 2 data, we interpreted the Ap stars showing no such variability over the 27-d duration of a TESS sector as being ssrAp star candidates. Here, we applied the same approach to TESS Cycle 3 and Cycle 4 observations of Ap stars. We show, however, that two issues that had not been fully appreciated until now may lead to spurious identification of ssrAp star candidates. On the one hand, a considerable fraction of the Ap stars in the existing lists turn out to have erroneous or dubious spectral classifications. On the other hand, the TESS data processing may remove part of the variability signal, especially for stars with moderately long periods ($20\ d After critical evaluation of these effects, we report the identification of 25 new ssrAp star candidates and of eight 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 sim 2 and sim 3\,kG 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. The next step to gain further understanding of the ssrAp stars will be to obtain high-resolution spectra of those for which such observations have not been made yet, to constrain their rotation velocities and their magnetic fields.
