Dettmann's truncated-correlation conjecture for periodic Lorentz processes

Let $\mathcal L$ be a lattice defining a periodic Lorentz process with phase space $M$, invariant measure $\mu$, free-flight time $\tau$, and flow $\Phi_t$. Let $f,g:M\to\mathbb R$ be H"older functions with zero mean with respect to $\mu$. Let $F(t)$ denote the free-path tail probability. Dettmann's truncated-correlation conjecture. As $t\to\infty$,

$$\int_{\{x\in M\mid\tau(x)<t\}}f(x)g(\Phi_t(x))\,d\mu=o(F(t)).$$

This dynamical conjecture asserts that correlations truncated to trajectories with free-flight time below $t$ are negligible compared with the free-path tail. Its status is not resolved in the supplied text.