Infinite Hex is arithmetic

Hex is a well known connection game in which two players attempt to connect opposite sides of the board by colored stones. In 2022, Hamkins and Leonessi introduced an infinite version, in which the goal is to construct a certain kind of two-way infinite path of adjacent stones. It was explicitly left open whether the winning condition is Borel. We prove that it is arithmetic, with complexity between $Σ^0_4$ and $Δ^0_5$.