De Palma–Trevisan's triangle inequality conjecture for the quantum Wasserstein divergence

Let $W_{2,\mathrm{DPT}}$ be the quantum Wasserstein divergence defined from the De Palma–Trevisan quantum Wasserstein distance by

$$W_{2,\mathrm{DPT}}(\rho,\sigma)=\sqrt{D(\rho,\sigma)^2-\frac 1 2D(\rho,\rho)^2-\frac 1 2 D(\sigma,\sigma)^2}.$$

Here $D(\rho,\sigma)$ is the original quadratic quantum Wasserstein distance associated with a given set of self-adjoint cost operators $x_1,\dots,x_d$. De Palma–Trevisan's conjecture. The map $W_{2,\mathrm{DPT}}$ satisfies the triangle inequality. This would show that the modified distance is a genuine metric, apart from possible degeneracy depending on the chosen cost operators. The triangle inequality was left open by De Palma and Trevisan.