Consider a number $n \in \mathbb{N}$, a finite field $\mathbb{F}_{p}$ (such that $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, and $\sqrt[3]{2} \notin \mathbb{F}_p$), and the elliptic $\mathbb{F}_{p}$-surface $$ \mathcal{E}_n\!: y^2 = x^3 + (t^{6n} + 1)^2. $$

Here I ask about infinite order $\mathbb{F}_{p}$-sections of $\mathcal{E}_1$. Is there such a section at least for some $\mathcal{E}_n$? In other words, the Mordell-Weil group of $\mathcal{E}_n$ is of positive rank for some $n \in \mathbb{N}$?

It seems to me that the answer is surprisingly negative, but I can not prove this. I only confirmed my conjecture by the computer algebra system Magma for small $p$ and $n$.