Young's shifted convolution conjecture for holomorphic cusp forms

Let $U(z)=y^{k/2}f(z)$, where $f$ is a holomorphic Hecke cusp form of weight $k$, with Fourier coefficients $\lambda(n)$ and first coefficient $\rho(1)$. For fixed $0<a\le y\le b$ and nonzero integer $m$, Young's shifted convolution conjecture for holomorphic forms.

$$\rho(1)^2\sum_{n\ge1}\frac{\lambda(n)\lambda(n+m)}{\sqrt{n(n+m)}}V_k(2\pi ny)V_k(2\pi(n+m)y)\ll_{a,b,\varepsilon}k^{-1/4+\varepsilon},$$

where equivalently

$$\int_0^1|U(x+iy)|^2e(mx)\,dx\ll_{a,b,\varepsilon}k^{-1/4+\varepsilon}.$$

More generally, for $M\le k^{1/2+\varepsilon}$,

$$\sum_{1\le|m|\le M}e(-m\alpha)\int_0^1|U(x+iy)|^2e(mx)\,dx\ll_{a,b,\varepsilon}k^{-1/4+\varepsilon}M^{1/2}.$$

These estimates are the holomorphic analogue of the Maass shifted-convolution hypotheses and are intended to control horocycle Fourier coefficients and restricted sup norms. The source does not report a proof, so the claims remain open.