The maximum number of cycles in a triangular-grid billiards system with a given perimeter

Given a grid polygon P in a grid of equilateral triangles, Defant and Jiradilok considered a billiards system where beams of light bounce around inside P. We study the relationship between the perimeter $\mathrm{perim}\left(P\right)$ of P and the number of different trajectories $\mathrm{cyc}\left(P\right)$ that the billiards system has. Resolving a conjecture of Defant and Jiradilok, we prove the sharp inequality $\mathrm{cyc}\left(P\right)\le \left(\mathrm{perim}\right(P)+2)/4$ and characterize the equality cases.