A challenger to elliptic billiards fails: String construction over convex polygons and the Birkhoff--Poritsky conjecture

We prove that the billiard claimed to be a possible counterexample to the Birkhoff-Poritsky conjecture is actually not a counterexample. We also show that for a billiard in a table obtained by the string construction over any convex polygon, this polygon is a core of the billiard dynamics.