The exact complexity of the infinite Post Correspondence Problem

In this short note, we give the exact complexity of the infinite Post Correspondence Problem, showing that it is Π10-complete. Surprisingly, it turns out that the infinite Post Correspondence Problem is not “more complex” than the Post Correspondence Problem, which is known to be Σ10-complete, but has the exact dual complexity. This gives an answer to a question of Simonnet [15].