Mean-square shifted convolution conjecture for non-holomorphic cusp-form coefficients

Let $t(n)$ denote the Fourier coefficients of the non-holomorphic cusp form considered in the paper, and let $m\sim M$ mean $M<m\leq 2M$. Let $N\geq 1$, $1\leq L\leq N$, and $1\leq F\ll N^{1-\varepsilon}$. Shifted convolution conjecture. One should have

$$\sum_{f\sim F}\sum_{n\sim N}\left|\sum_{l\sim L}t(n+l)t(n+l+f)\right|^{2}\ll N^{2+\varepsilon}+N^{1+\varepsilon}LF.$$

This is the conjectured analogue of the established mean-square estimate for the divisor function and the corresponding result for holomorphic cusp forms. It is proposed because the spectral decomposition for non-holomorphic cusp forms contains an additional arithmetic correction term that obstructs the proof; the source gives no resolution of the conjecture.