In cases where both components of a binary system show oscillations, asteroseismology has been proposed as a method to identify the system. For KIC 2568888, observed with $Kepler$, we detect oscillation modes for two red giants in a single power density spectrum. Through an asteroseismic study we investigate if the stars have similar properties, which could be an indication that they are physically bound into a binary system. While one star lies on the red giant branch (RGB), the other, more evolved star, is either a RGB or asymptotic-giant-branch star. We found similar ages for the red giants and a mass ratio close to 1. Based on these asteroseismic results we propose KIC 2568888 as a rare candidate binary system ($sim 0.1%$ chance). However, when combining the asteroseismic data with ground-based $BVI$ photometry we estimated different distances for the stars, which we cross-checked with $Gaia$ DR2. From $Gaia$ we obtained for one object a distance between and broadly consistent with the distances from $BVI$ photometry. For the other object we have a negative parallax with a not yet reliable $Gaia$ distance solution. The derived distances challenge a binary interpretation and may either point to a triple system, which could explain the visible magnitudes, or, to a rare chance alignment ($sim 0.05%$ chance based on stellar magnitudes). This probability would even be smaller, if calculated for close pairs of stars with a mass ratio close to unity in addition to similar magnitudes, which may indeed indicate that a binary scenario is more favourable.
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