Existence of Probability Measures With Given Marginals and moment constraints

Question

let $\mu$ and $\nu$ be two probability measures on $\mathbb{R}^d$, my question under which conditions is there a coupling measure between $\mu$ and $\nu$ that satisfies some moment constraints. for example $\pi\in\Gamma(\mu,\nu)$ s/c $$\int_{\mathbb{R}^d\times\mathbb{R}^d}<x,y>d\pi(x,y)=0$$ beside the work of Strassen, V. The Existence of Probability Measures with Given Marginals, where he studied a more general form of my question i didn't find any paper related to moment. Can someone please recommend any sources that are related to my question.