Precise radial velocities of giant stars -- XVII. Distinguishing planets from intrinsically induced radial velocity signals in evolved stars

From a long-term Doppler monitoring campaign of 373 giant stars, we have identified ten giants with periodic radial velocity variations that are challenging to associate with planets. Similar cases in the literature are attributed to poorly understood intrinsic processes. Our goal is to confirm or refute the presence of planets around these ten evolved stars. Additionally, we evaluate the reliability and sensitivity of planet-confirmation metrics when applied to giant stars and present cases of intrinsically induced radial velocity variations, aiming to enhance the physical understanding of the phenomenon. We combined 25 years of radial velocity data from the Hamilton/Lick, SONG, and CARMENES spectrographs. To assess consistency with Keplerian models, we examined the residuals and tracked changes in statistical significance as new data were incorporated. Additionally, we compared radial velocity amplitudes across optical and infrared wavelengths, searched for periodic variations of activity indicators, and examined their correlations with radial velocities. Seven of the ten giants exhibit intrinsically induced radial velocity variations. The strongest arguments against planets orbiting the giants are guided by long-term radial velocity monitoring that detects changing periodicity on long timescales or detects systematics close to the original period in the radial velocity residuals. While activity indicators offer some support, their signals are generally weak. Comparing optical and infrared radial velocity amplitudes also proves insufficient for confirming or refuting planets. We find HIP 64823 remains a promising candidate for hosting a giant exoplanet with orbital period $P\sim$ 7.75 yr. For two stars, the evidence remains inconclusive. Long-term radial velocity monitoring is essential for distinguishing planetary companions from intrinsic variations in evolved stars.