Young's square-root shifted convolution conjecture for Maass forms

Let $U(z)$ be a Hecke–Maass cusp form with Laplace eigenvalue $1/4+T^2$, Fourier coefficients $\lambda(n)$ and first coefficient $\rho(1)$. For fixed $0<a\le y\le b$, $\alpha\in\mathbb{R}$, and $1\le M\le T^{1+\varepsilon}$, Young's square-root shifted convolution conjecture.

$$|\rho(1)|^2\sum_{1\le|m|\le M}e(m\alpha)\sum_{n\in\mathbb{Z}}\lambda(n)\lambda(n+m)K_{iT}(2\pi|n|y)K_{iT}(2\pi|m+n|y)\ll_{a,b,\varepsilon}T^{-1/2+\varepsilon}M^{1/2}.$$

Equivalently, uniformly in $\alpha\in\mathbb{R}$,

$$\sum_{1\le|m|\le M}\int_0^1|U(x+iy)|^2e(m(\alpha-x))\,dx\ll_{a,b,\varepsilon}T^{-1/2+\varepsilon}M^{1/2}.$$

This stronger hypothesis supplies cancellation across the shift $m$ and would imply the conjectural compact-set sup-norm bound $U(z)\ll T^\varepsilon$.