  Quantum Wasserstein divergences are modified versions of quantum Wasserstein
distances defined by channels, and they are conjectured to be genuine metrics
on quantum state spaces by De Palma and Trevisan. We prove triangle inequality
for quantum Wasserstein divergences for every quantum system described by a
separable Hilbert space and any quadratic cost operator under the assumption
that a particular state involved is pure, and all the states have finite
energy. We also provide strong numerical evidence suggesting that the triangle
inequality holds in general, for an arbitrary choice of states.
