The almost-everywhere entry conjecture for the region U

Let $\triangle$ be the simplex for the $\mathbb{R}^3(e,13,e)$ TRIP map, and let

$$U=\{(x,y,z)\in\triangle:y>z\}.$$

Almost-everywhere entry conjecture. For Lebesgue-almost every point in $\triangle$, the region $U$ can be reached after finitely many iterations of the $\mathbb{R}^3(e,13,e)$ TRIP map.

If true, this would extend the reduction to a two-dimensional TRIP map from points initially in $U$ to almost every point of the simplex. The source reports computational evidence and leaves the proof to future work.