Consider a natural number $n$, a finite field $\mathbb{F}_{p}$ (such that $p$ is prime, $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, 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 asked about infinite order $\mathbb{F}_{p}$-sections of $\mathcal{E}_1$. Is there such a section of $\mathcal{E}_n$ at least for some natural $n$? In other words, is the Mordell-Weil group of $\mathcal{E}_n$ of positive rank for some $n$?

The answer seems to be negative, but I cannot prove this. I have only confirmed my conjecture by the computer algebra system Magma for small $p$ and $n$.