Blomer–Khan–Young fourth-moment conjecture for holomorphic cusp forms

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\operatorname{SL}_2(\mathbb{Z})$. For $z=x+iy$ in the upper half-plane $\mathbb{H}$, define

$$F(z)=y^{k/2}f(z).$$

For $1\le p<\infty$, define

$$\lVert F\rVert_p=\left(\int_{\operatorname{SL}_2(\mathbb{Z})\backslash\mathbb{H}}|F(z)|^p\frac{dx\,dy}{y^2}\right)^{1/p}.$$

Blomer–Khan–Young's fourth-moment conjecture. Under the normalization

$$\frac{3}{\pi}\lVert F\rVert_2^2=1,$$

we have

$$\frac{3}{\pi}\lVert F\rVert_4^4\sim 2$$

as $k\to\infty$. This conjecture concerns the expected fourth moment of normalized holomorphic cusp forms and is motivated by the random wave model. The source presents it as an earlier conjecture and does not state that it has been proved or disproved.