Davis–Lelièvre's local-global conjecture for asymmetric pentagonal billiard periods

A combinatorial period length is the number of line segments in a periodic trajectory on the regular pentagon; a trajectory is asymmetric when it does not have 5-fold rotational symmetry. Davis–Lelièvre's local-global conjecture. Every sufficiently large even number arises as an asymmetric combinatorial period length on the regular pentagon. More precisely, every positive even integer except $2$, $12$, $14$ and $18$ occurs. This conjecture was motivated by extensive numerical analysis of closed trajectories and concerns the complete set of asymmetric combinatorial period lengths.