If you say map instead of embedding and measure area with multiplicity, then the problem become more interesting.

The following analog of Nash--Kuiper theorem was proved by Gromov in his "Partial differential relations":

Let $f\colon M\to N$ be a short map between Riemannian manifolds of the same dimenssion. Then there is arbitrarily close length-preserving map $f_\varepsilon\colon M\to N$.

The provided map is evidently a maximizer. It is not smooth and typically has creases at everywhere dense subset, but one can smooth it keeping the area nearly the same.

If you say map instead of embedding and measure area with multiplicity, then the problem becomes more interesting.

The following analog of Nash--Kuiper theorem was proved by Gromov in his "Partial differential relations":

Let $f\colon M\to N$ be a short map between Riemannian manifolds of the same dimenssion. Then there is arbitrarily close length-preserving map $f_\varepsilon\colon M\to N$.

The provided map is evidently a maximizer. It is not smooth and typically has creases at everywhere dense subset, but one can smooth it keeping the area nearly the same.