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The Close AGN Reference Survey (CARS): No obvious signature of AGN feedback on star formation, but subtle trends

[Abridged] Active Galactic Nuclei (AGN) are thought to be responsible for the suppression of star formation in massive ~10$^{10}$ M$_\odot$ galaxies. While this process is a key feature in numerical simulations, it is not yet unambiguously confirmed in observational studies. Characterization of the star formation rate (SFR) in AGN host galaxies is challenging as AGN light contaminates most SFR tracers. We aim to obtain and compare SFR estimates from different tracers for AGN host galaxies in the Close AGN Reference Survey (CARS) to provide new observational insights. We construct integrated panchromatic spectral energy distributions (SED) to measure the FIR luminosity as a tracer for the recent (< 100 Myr) SFR. In addition, we use integral-field unit observation of the CARS targets to employ the H$α$ luminosity decontaminated by AGN excitation as a proxy for the current (< 5 Myr) SFR. We find that significant differences in specific SFR of the AGN host galaxies as compared with the larger galaxy population disappear once cold gas mass, in addition to stellar mass, is used to predict the SFR. We identify individual galaxies with a significant difference in their SFR which can be related to a recent enhancement or decline in their SFR history that might be related to various processes including interactions, gas consumption, outflows and AGN feedback. AGN can occur in various stages of galaxy evolution which makes it difficult to relate the SFR solely to the impact of the AGN. We do not find any strong evidence for global positive or negative AGN feedback in the CARS sample. However, there is tentative evidence that 1) the relative orientation of the AGN engine with respect to the host galaxies might alter the efficiency of AGN feedback and 2) the recent SFH is an additional tool to identify rapid changes in galaxy growth driven by the AGN or other processes.