Dettmann's maximal-horizon asymptotic conjecture for periodic Lorentz processes

Let $\mathcal L$ be a lattice defining a periodic Lorentz process, let $F(t)$ be the free-path tail probability, and let $\mathbb H$ denote the set of maximal non-incipient horizons. For each horizon $H$, let $F_H(t)$ be the probability that a free path remains in $H$ throughout the time interval $[0,t]$. Dettmann's maximal-horizon conjecture. If the process has at least one non-incipient maximal horizon, then, as $t\to\infty$,

$$F(t)\sim\sum_{H\in\mathbb H}F_H(t).$$

This conjecture predicts that the tail of the free-path distribution is asymptotically determined by the maximal non-incipient horizons. Its status is not resolved in the supplied text.