Sign-preserving transport cost for finite signed measures

Assume $d\ge 1$ and the Axiom of Choice when invoking Jordan decomposition of a signed measure into unique mutually singular positive parts. Let $\omega ,\nu$ be finite signed Borel measures on ${\mathbb{R}}^d$ whose total variations have finite first moments, and use the Euclidean norm from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Write their Jordan parts as ${\omega }^{+},{\omega }^{-}$ and ${\nu }^{+},{\nu }^{-}$. If the positive-part masses agree and the negative-part masses agree, define using finite nonnegative Borel measures ${\pi }^{\pm }$ on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ whose first and second marginals are respectively the corresponding positive or negative Jordan parts: $C_{\pm }(\omega ,\nu ):={\inf }_{{\pi }^{+},{\pi }^{-}}(\int \mathrm{\mid }x-y\mathrm{\mid }d{\pi }^{+}(x,y)+\int \mathrm{\mid }x-y\mathrm{\mid }d{\pi }^{-}(x,y)),$ where ${\pi }^{+}$ ranges over finite nonnegative couplings of ${\omega }^{+}$ and ${\nu }^{+}$, and ${\pi }^{-}$ over finite nonnegative couplings of ${\omega }^{-}$ and ${\nu }^{-}$. The infimum is over both sign couplings independently. This cost is not the probability $W_1$.