  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.
