Nondegeneracy of the relative p-Wasserstein distance for p>1

Establish whether the relative p-Wasserstein distance W_p defines a genuine metric on the space M of p-finite relative Radon measures for p>1; equivalently, prove that for any μ,ν ∈ M, W_p(μ,ν)=0 implies μ=ν.

Background

The authors show that W_p is a pseudometric on M, proving reflexivity, symmetry, and the triangle inequality, and that the Dirac embedding (X,d_p)→(M,W_p) is isometric. However, for p>1 they do not show that W_p is nondegenerate on M (i.e., that zero distance implies equality of measures).

Clarifying whether W_p is a metric for p>1 would strengthen the theoretical foundations of relative optimal transport, aligning it with the classical case where the p-Wasserstein distance is a metric on spaces of finite measures with equal mass.