Lamberti et al.'s metric conjecture for the quantum Jensen–Shannon divergence

Let $\mathbb{P}_d$ denote the cone of $d\times d$ positive definite matrices, and for $X,Y\in\mathbb{P}_d$ let $\operatorname{QJSD}(X,Y)$ be the quantum Jensen–Shannon divergence.

Lamberti et al.'s conjecture. $\operatorname{QJSD}^{1/2}$ is a metric on $\mathbb{P}_d$.

The conjecture asks whether the square root of the quantum Jensen–Shannon divergence satisfies the triangle inequality for general quantum states. The source reports that the question was open apart from parallel work that provides a proof, so its resolution should be checked against that work.