Suppose that $\mathcal{C}$ and $\mathcal{D}$ are subsets of $L^2(X,\Sigma,\mu)\cap L^{\infty}(X,\Sigma,\mu)$, where $\mu$ is a finite-measure on $(X,\Sigma)$.  Let $F:L^2(X,\Sigma,\mu)\times L^2(X,\Sigma,\mu)\rightarrow \mathbb{R}$ be of the form
$$
F(f,g)\triangleq \int_{x \in X} G(f(x),g(x)) \mu(dx),
$$
for some continuous map $G:\mathbb{R}^2 \rightarrow \mathbb{R}$.  

Assume that:

 - The map
$y\mapsto \operatorname{argmin}_{f \in \mathcal{D}}F(g,f)$ is single-valued on $\mathcal{D}$ and (depends on the choice of $g$),
 - The assumptions of [Sion's Minimax Theorem][1] hold.

Does this guarantee that
$$
\operatorname{argmin}_{g \in \mathcal{C}}\operatorname{sup}_{f \in \mathcal{D}}
F(g,f)
=
\operatorname{esssup}_{g \in \mathcal{D}}
\operatorname{argmin}_{f \in \mathcal{C}}F(g,f)
?
$$

If not, what additional assumptions are needed?

  [1]: https://en.wikipedia.org/wiki/Sion%27s_minimax_theorem