Ulmer (and then other authors) showed existence of elliptic curves $E$ over the function field $\mathbb{F}_{p}(t)$ (where $p$ is a prime) with huge Mordell-Weil ranks (i.e., depending on $p$). The curves $E$ are non-isotrivial, that is, their $j$-invariants $j_E(t)$ are non-constant rational functions. However, for some $j \in \mathbb{F}_{p}$ there are values $t_0 \in \mathbb{F}_{p}$ such that $j_E(t_0) = j$.
Is it true that for each (quite large) prime $p$ and for each (ordinary) $j$-invariant $j \in \mathbb{F}_{p}$ there is an elliptic curve $E$ over $\mathbb{F}_{p}(t)$ with huge Mordell-Weil rank and there is a value $t_0 \in \mathbb{F}_{p}$ such that $j_E(t_0) = j$?
