  Quantum optical systems are typically affected by two types of noise: photon
loss and dephasing. Despite extensive research on each noise process
individually, a comprehensive understanding of their combined effect is still
lacking. A crucial problem lies in determining the values of loss and dephasing
for which the resulting loss-dephasing channel is anti-degradable, implying the
absence of codes capable of correcting its effect or, alternatively, capable of
enabling quantum communication. A conjecture in [Quantum 6, 821 (2022)]
suggested that the bosonic loss-dephasing channel is anti-degradable if and
only if the loss is above $50\%$. In this paper we refute this conjecture,
specifically proving that for any value of the loss, if the dephasing is above
a critical value, then the bosonic loss-dephasing channel is anti-degradable.
While our result identifies a large parameter region where quantum
communication is not possible, we also prove that if two-way classical
communication is available, then quantum communication -- and thus quantum key
distribution -- is always achievable, even for high values of loss and
dephasing.
