I'm trying to understand a result in optimal transport. Specifically, consider the Distributionally Robust Optimization problem given by $$\sup \left\{\int f d\mu : d_{c}(\mu, \nu) \leq \delta\right\}$$ where $\delta>0$, $f$ is integrable and lower semicontinuous, measures $\mu, \nu$ live in a Polish space $P(S)$ and $d_{c} = \inf_{\pi\in\Pi(\mu, \nu)}\int c(x,y)\pi(dx, dy)$ with $\Pi$ the set of couplings with marginals $\mu, \nu$.
In Chapter 4 of Villani's Optimal Transport: Old and New it is claimed that if the cost is nonnegative then an optimal coupling exists. Therefore in this other paper https://arxiv.org/pdf/1604.01446.pdf it is stated that we can rewrite the former problem as: $$ \sup \left\{\int f d\pi : \pi \in \bigcup_{\nu\in P(S)} \Pi(\mu, \nu), \int cd\pi \leq\delta\right\}.$$ Can someone give me the intuition why one can discard the optimality requirement of the coupling and consider also the integral with respect to the measure $\pi$ instead of $\mu$?