Equality of the two relative p-Wasserstein formulations for p>1

Determine whether, for every metric pair (X,d,A) with (X,d) complete and separable and A nonempty, and for all relative Borel measures μ,ν on X, the two definitions of the relative p-Wasserstein distance coincide for p>1; specifically, show that (inf_{π∈Π(μ,ν)} π(d_p^p))^{1/p} equals (inf_{ε>0} inf_{π∈Π_ε(μ,ν)} π(d_p^p))^{1/p}, where d_p(x,y)=min(d(x,y),((d_A(x))^p+(d_A(y))^p)^{1/p}) and Π_ε(μ,ν) consists of couplings that are trivial extensions of couplings between the ε-offset restrictions μ_ε and ν_ε.

Background

The paper introduces two closely related formulations of the relative p-Wasserstein distance on metric pairs (X,d,A). The first, denoted \hat{W}_p, takes the infimum over all couplings of the integral of d_p^p; the second, denoted W_p, takes the infimum over ε>0 of couplings that arise from couplings of the ε-offset restricted measures, extended trivially via the reservoir A.

For p=1 the authors prove that the two notions coincide (\hat{W}_1=W_1). For p>1 the equivalence is not established, and resolving whether the two formulations agree would clarify the canonical choice of relative p-Wasserstein distance in the unbalanced/relative transport framework developed in the paper.