Mean-energy equality conjecture for Brody curves in the projective line

For a Brody curve $f$ in $\mathbb{C} P^1$, let $e(f)$ be its mean energy, and let $e(\mathbb{C} P^1)$ be the supremum of $e(f)$ over all Brody curves in $\mathbb{C} P^1$. Let $e(\mathbb{C} P^1)_{ell}$ be the supremum over elliptic Brody curves. Mean-energy equality conjecture.

$$e(\mathbb{C} P^1)_{ell}=e(\mathbb{C} P^1).$$

The equality would say that elliptic Brody curves attain the same supremal mean energy as arbitrary Brody curves in the projective line. Together with the mean-dimension bounds, it would identify the lower and upper estimates in dimension one; the source proposes it as a conjecture without giving a resolution.