Young's logarithmic geodesic $L^2$ conjecture for Maass forms

Let $u_j$ be an even Hecke–Maass cusp form with spectral parameter $t_j$, so its Laplace eigenvalue is $1/4+t_j^2$. Young's logarithmic geodesic $L^2$ conjecture.

$$\int_0^\infty|u_j(iy)|^2\frac{dy}{y}\sim2\log\left(\frac14+t_j^2\right)$$

as $t_j\to\infty$. This is the $\alpha=0$ specialization of the noncompact weighted geodesic conjecture, after resolving the Eisenstein-series singularity. The source presents it as a prediction and motivates it through the localization of the Maass $K$-Bessel functions.