Generic equidistribution conjecture for rays in positive-characteristic homogeneous dynamics

Let $X_\infty$ be the homogeneous space in question, let $\Omega$ be the space of rays, and for each $\xi\in\Omega$ let $x_n^\xi$ be the associated sequence of points with periodic measure $\mu_{x_n^\xi}$. The space $\Omega$ carries its natural probability measure.

Generic equidistribution conjecture. For almost every $\xi\in\Omega$, the averages

$$\frac{1}{N+1}\sum_{n=0}^N\mu_{x_n^\xi}$$

converge to the homogeneous probability measure on $X_\infty$.

This conjecture predicts that uniform escape of mass, observed for rational rays, is nongeneric; it is presented as an open belief motivated by the contrast between positive- and zero-characteristic dynamics.