Assume that $M$ is a non-standard model of complete arithmetic, i.e. of the theory $Th(\mathbb{N})$. Suppose that $R$ and $S$ are proper cuts of $M$. (With a cut, I mean a subset of the universe of $M$ which is closed under successor and which is downward closed.) Let $\varphi(x,y)$ be a formula in the language of arithmetic, possibly with parameters from $M$. My question is whether the following variant of the overspill principle holds:

If for all $s \in S$ and all $r \in R$, it holds that $M\vDash \varphi(s,r)$, then there is an element $s_0 > S$ and an element $r_0 > R$ with $M \vDash \varphi(s_0,r_0)$.

It will also be helpful for me to learn about some "interesting" special cases where the above statement may hold (e.g. restrictions on the cuts $R$ and $S$, restrictions on the formula $\varphi$, ...).