A question on periodic orbits for billiards

In section 2.9 of the book “Notes on Dynamical Systems” by Moser and Zehnder, it is proved the following theorem as a corollary of the PoincarĂ©-Birkhoff fixed point theorem:

Theorem: On a strictly convex billiard table, there exist infinitely many distinct periodic orbits.

Is the result, or a weaker version of it, still true for convex tables?

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