= Solution
For $p\in\mathcal U^*$, the definition of the <convex conjugate> gives
$$
\begin{aligned}
(E\mathbin\square F)^*(p)
&=\sup_u\sup_v\{\langle p,u\rangle-E(v)-F(u-v)\}\\
&=\sup_{v,w}\{\langle p,v\rangle-E(v)+\langle p,w\rangle-F(w)\}\\
&=E^*(p)+F^*(p),
\end{aligned}
$$
where the substitution $w=u-v$ is a bijection of the independent pairs. Therefore
$$
\boxed{(E\mathbin\square F)^*=E^*+F^*.}
$$
This <infimal convolution> identity does not need <convexity> of $E,F$ or attainment of the inner <infimum>. The assumptions ensure that both functions have nonempty <effective domains>; their conjugates never take $-\infty$, so the separated sum is well defined, allowing $+\infty$.
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