On the size of a linear combination of two linear recurrence sequences over function fields

Let $G_n$ and $H_m$ be two non-degenerate linear recurrence sequences defined over a function field $F$ in one variable over $\mathbb{C}$, and let $\mu$ be a valuation on $F$. We prove that under suitable conditions there are effectively computable constants $c_1$ and $C'$ such that the bound \begin{equation*} \mu(G_n - H_m) \leq \mu(G_n) + C' \end{equation*} holds for $\max \{n,m\} > c_1$.