De Palma–Trevisan conjecture on quantum Wasserstein divergences

Let $\mathcal{H}$ be a Hilbert space, let $\mathcal{A}=\{A_1,\dots,A_k\}$ be a finite collection of observables on $\mathcal{H}$, and let $D_{\mathcal{A}}$ be the associated quantum optimal transport distance. For states $\rho,\omega\in\mathcal{S}(\mathcal{H})$ with finite energy, define

$$d_{\mathcal{A}}(\rho,\omega):=\sqrt{D_{\mathcal{A}}^2(\rho,\omega)-\frac{1}{2}\left(D_{\mathcal{A}}^2(\rho,\rho)+D_{\mathcal{A}}^2(\omega,\omega)\right)}.$$

De Palma–Trevisan conjecture. Up to non-degeneracy assumptions on the $A_j$ ensuring definiteness, the function $d_{\mathcal{A}}$ is a true metric on the set of states on $\mathcal{H}$ with finite energy relative to $\mathcal{A}$; in particular, $d_{\mathcal{A}}(\rho,\omega)=0$ only if $\rho=\omega$. Quantum Wasserstein divergences were introduced as modifications of quantum Wasserstein distances precisely to remove the possible positive self-distance. The conjecture asserts that this modification also satisfies the triangle inequality, while definiteness requires the stated additional assumptions.