  In a recent quantum key distribution experiment, high-dimensional protocols
were used to show an improved noise resistance over a 10.2 km free-space
channel. One of the unresolved questions in this context is whether the
communicating parties actually shared genuine high-dimensional entanglement. In
this letter we introduce an improved discretisation and entanglement
certification scheme for high-dimensional time-bin setups and apply it to the
data obtained during the experiment. Our analysis answers the aforementioned
question affirmatively and thus the experiment constitutes the first
transmission of genuine high-dimensional entanglement in a single degree of
freedom over a long-range free-space channel.
