CFKRS conjecture for the fourth moment of holomorphic cusp forms

Let $f$ be a holomorphic cusp form of weight $k$ for the full modular group, and write $\bBbb H$ for the upper half-plane and $\Gamma$ for the full modular group. Normalize $f$ by

$$\int_{\Gamma \backslash \mathbb{H}} |f(z)|^2 y^k \frac{3}{\pi} \frac{dx\,dy}{y^2}=1.$$

CFKRS conjecture. As $k\to\infty$,

$$\int_{\Gamma \backslash \mathbb{H}} |f(z)|^4 y^{2k} \frac{3}{\pi} \frac{dx\,dy}{y^2}=2+o(1).$$

This predicts the true asymptotic size of the fourth norm of a normalized holomorphic cusp form; the paper establishes an upper bound of order $k^{1/3+\varepsilon}$ for the fourth power, while the stated asymptotic remains unproved here.