Young's horocycle QUE conjecture for restricted eigenfunctions

Let $U(z)$ be either $y^{k/2}f(z)$ or $u_j(z)$, and let $\psi:\mathbb{Z}\backslash\mathbb{R}\to\mathbb{R}$ be smooth. For fixed $y>0$, Young's horocycle QUE conjecture.

$$\int_0^1\psi(x)|U(z)|^2\,dx\sim\int_0^1\psi(x)\,dx$$

as the weight or eigenvalue of $U$ tends to infinity. This is the horocycle analogue of geodesic restriction QUE. It is stated as a conjectural extension of ambient quantum ergodicity to fixed horocycles.