Nonexistence of higher-degree smooth absolute minima in the standard contact 3-sphere model

Let $M=\mathbb{S}^3$ and $N=S^3$ be equipped with the standard contact structure. Consider smooth maps from $M$ to $N$, with degree defined in the usual way, and the energy functional $E$ given by

$$E=\text{the energy in }.$$

Nonexistence conjecture. There are no smooth absolute minima of $E$ of degree greater than $1$, irrespective of the coupling constant.

This question arises from the rigidity of strong Beltrami maps between standard contact 3-spheres. Establishing the claim would rule out higher-degree smooth absolute-energy minimizers in this model, while the surrounding discussion leaves the existence of such minimizers as an open problem.