Let $X, Y$ be Polish spaces with probability measures $\mu, \nu$, respectively; and let $\gamma$ be a finite measure on the product $X \times Y$. We consider the maximization problem $$ \sup_{\varphi,\psi} \bigg\{\int_{X} \varphi \ d\mu + \int_{Y} \psi \ d\nu - \int_{X \times Y} \big( \varphi(x) + \psi(y) \big) \ d\gamma\bigg\} $$ over the set of all bounded and continuous functions $\varphi(x),\psi(y)$.
Why can this maximization problem always attain the value $+\infty$ for every initial choices of $\mu,\nu,\gamma$, if we exclude the case where $\gamma$ has marginals $\mu$ and $\nu$?
Let $\mu'$ and $\nu'$ be the margins of $\gamma$. At least one of $\mu\ne\mu'$ and $\nu\ne\nu'$ must hold. Your maximization problem becomes: maximize $$\int_X \varphi(d\mu-d\mu') +\int_Y\psi(d\nu-d\nu').$$ Assume without loss of generality that $\mu\ne\mu'$, let the signed measure $\sigma=\mu-\mu'.$ Then your maximization problem is lower bounded by the restricted maximization problem, of determining $$\sup_\varphi \int_{X}\varphi d\sigma.$$ Since $\sigma$ is not the zero measure, there exists a continuous bounded $f$ such that $\int_X f\sigma = 1$. Now it is clear that by taking $\varphi$ to be large scalar multiples of $f$, you see the supremum is $\infty$.