  A strong converse theorem for channel capacity establishes that the error
probability in any communication scheme for a given channel necessarily tends
to one if the rate of communication exceeds the channel's capacity.
Establishing such a theorem for the quantum capacity of degradable channels has
been an elusive task, with the strongest progress so far being a so-called
"pretty strong converse". In this work, Morgan and Winter proved that the
quantum error of any quantum communication scheme for a given degradable
channel converges to a value larger than $1/\sqrt{2}$ in the limit of many
channel uses if the quantum rate of communication exceeds the channel's quantum
capacity. The present paper establishes a theorem that is a counterpart to this
"pretty strong converse". We prove that the large fraction of codes having a
rate exceeding the erasure channel's quantum capacity have a quantum error
tending to one in the limit of many channel uses. Thus, our work adds to the
body of evidence that a fully strong converse theorem should hold for the
quantum capacity of the erasure channel. As a side result, we prove that the
classical capacity of the quantum erasure channel obeys the strong converse
property.
