We show that the conjugacy problem is undecidable in the Brin-Thompson group $2V$. In fact, given two elements $g$, $h$, one cannot recursively separate the following properties:

1. $g$ and $h$ are of finite order, and are conjugate by an involution,
2. $g$ and $h$ are of infinite order, and are not conjugate.

The proof is based on the Kari-Ollinger proof of undecidability of periodicity in reversible Turing machines, and some ideas from symbolic dynamics and one-head machine dynamics.