  Let $S$ be a closed surface of genus at least $2$. For each maximal
representation $\rho: \pi_1(S)\rightarrow\mathsf{Sp}(4,\mathbb{R})$ in one of
the $2g-3$ exceptional connected components, we prove there is a unique
conformal structure on the surface in which the corresponding equivariant
harmonic map to the symmetric space $\mathsf{Sp}(4,\mathbb{R})/\mathsf{U}(2)$
is a minimal immersion. Using a Higgs bundle parameterization of these
components, we give a mapping class group invariant parameterization of such
components as fiber bundles over Teichm\"uller space. Unlike Labourie's recent
results on Hitchin components, these bundles are not vector bundles.
