Let $A$ and $B$ be objects in a topos $\mathcal{E}$ with natural numbers object $\mathbb{N}$. If there exists a monomorphism $m: \mathbb{N}\rightarrow A\times B$, is it necessarily the case that there is either a monomorphism from $\mathbb{N}$ to $A$ or a monomorphism from $\mathbb{N}$ to $B$?

This is clearly true when $\mathcal{E}$ is Boolean, although the proof I've seen is more involved than one might expect. It freely uses the Boolean assumption in several places.