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

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