Photometric follow-up of the 20 Myr-old multi-planet host star V1298~Tau with CHEOPS and ground-based telescopes

V1298 Tau hosts at least four planets. Since its discovery, this system has been a target of intensive photometric and spectroscopic monitoring. The characterisation of its architecture and planets' fundamental properties turned out to be very challenging so far. The determination of the orbital ephemeris of the outermost planet V1298 Tau $e$ remains an open question. Only two transits have been detected so far by $Kepler/K2$ and TESS, allowing for a grid of reference periods to be tested with new observations, without excluding the possibility of transit timing variations. Observing a third transit would allow to better constrain the orbital period, and would also help determining an accurate radius of V1298 Tau $e$ because the former transits showed different depths. We observed V1298 Tau with the CHEOPS space telescope to search for a third transit of planet $e$ within observing windows that have been selected in order to test three of the shortest predicted orbital periods. We also collected ground-based observations to verify the result found with CHEOPS. We reanalysed $Kepler/K2$ and TESS light curves to test how the results derived from these data are affected by alternative photometric extraction and detrending methods. We report the detection with CHEOPS of a transit that could be attributed to V1298 Tau $e$. If so, that result implies that the orbital period calculated from fitting a linear ephemeris to the three available transits is close to $\sim45$ days. Results from the ground-based follow-up marginally support this possibility. We found that $\textit{i}$) the transit observed by CHEOPS has a longer duration compared to that of the transits observed by $Kepler/K2$ and TESS; $\textit{ii}$) the transit observed by TESS is $>30\%$ deeper than that of $Kepler/K2$ and CHEOPS, and deeper than the measurement previously reported in the literature, according to our reanalysis.