  We prove an analog of Bryant's duality theorem for a four dimensional
Willmore energy $\mathcal{E}_{GR}$ obtained by Graham-Reichert and Zhang. We
show that for an immersion $\Phi$ from a four dimensional compact manifold
without boundary $\Sigma$ into $\mathbb{R}^5$, the energy
$\mathcal{E}_{GR}(\Phi)$ is equal to two energies on its conformal Gauss map
$Y$. One defined only in terms of the image of $Y$, which is the analog of the
area functional for Willmore surfaces, and an other one defined on maps from
$\Sigma$ into the De Sitter space $\mathbb{S}^{5,1}$, which is the analog of
the Dirichlet energy for Willmore surfaces. We prove that even when restricted
to immersions of a given topological manifold $\Sigma^4$, $\mathcal{E}_{GR}$ is
never bounded from below on the set of immersions from $\Sigma$ into
$\mathbb{R}^5$. We exhibit a second conformally invariant energy
$\mathcal{E}_P$ which is bounded from below and whose construction is closer to
the two dimensional Willmore energy.
