Finite-rank conjecture for grouplike types in $\mathbb{Q}$ACFA

Let $L_{\mathbb{Q}}$ be the language of fields with a $(\mathbb{Q},+)$-action, let $L_1$ be its reduct containing the action of $1$, and let $p$ be an $L_{\mathbb{Q}}$-type with reduct $p_1$ to $L_1$. A type is grouplike when it is nonorthogonal to a generic type of a minimal modular group, and rank is measured in the corresponding theory.

Finite-rank conjecture. Suppose that $p_1$ is minimal in the sense of $L_1$ and ACFA. If $p_1$ is grouplike in the sense of $L_1$ and ACFA, then $p$ has finite rank in the sense of $\mathbb{Q}$ACFA.

This concerns whether passage from the ACFA reduct to the full theory with a rational action can cause the rank of a grouplike type to become infinite. The surrounding discussion gives examples where rank can increase, but presents finite rank as the expected general bound.