  Superconducting islands (SIs) on hybrid nanowires are routinely used to
search for signatures of Majorana zero modes (MZMs). Whereas the coupling of a
quantum dot (QD) to a large grounded superconducting reservoir produces
discrete Yu-Shiba-Rusinov subgap states with a perfect positive-negative bias
symmetry, it has been recently predicted that the subgap spectrum resulting
from the coupling of a QD to a small electrically isolated SI is in general not
symmetric. It is furthermore affected by the even-odd parity alternation in the
occupation of the SI enforced by Coulomb blockade due to large charging energy.
By performing differential conductance measurements on a QD-SI device and with
support of theoretical modelling, we show that the subgap states indeed lack
symmetry of positive and negative bias positions and that, in particular
instances, they even lack continuity. For odd parity of the SI in the ground
state and weak QD-SI hybridisation, we observe a bias-polarity dependent
current blockade in the stability diagram. At finite magnetic field and for
strong QD-SI hybridisation, zero-bias crossings of the states can mimic
signatures of electron teleportation via two well-separated MZMs at the ends of
the SI.
