Dettmann's incipient-principal-horizon tail conjecture for periodic Lorentz processes

Let $\mathcal L$ be a lattice defining a periodic Lorentz process, let $d$ be the dimension, and let $F(t)$ be the free-path tail probability. A principal horizon has the highest possible dimension, and a horizon is incipient when its basis has measure zero in the relevant transverse dimension. Dettmann's incipient-principal-horizon conjecture. If the process has an incipient but no actual principal horizon, then, as $t\to\infty$,

$$F(t)\asymp\begin{cases}\nt^{-2},&d<6,\nt^{-2}\log t,&d=6,\nt^{-\alpha_d},\quad 1<\alpha_d<2,&d>6. \end{cases}$$

This geometric conjecture gives dimension-dependent tail asymptotics in the absence of an actual principal horizon. Its status is not resolved in the supplied text.