Asymptotic norm conjecture for restricted Saito–Kurokawa lifts

Let $f$ be a cusp form of weight $k$ whose Saito–Kurokawa lift is $F_f$, and let $N(F_f)$ denote the normalized norm of the restriction of $F_f$ to $z=0$ as defined in the paper.

Asymptotic norm conjecture. As $k$ tends to infinity,

$$N(F_f)\sim 2.$$

This is presented as a refinement beyond the bound $N(F_f)\ll k\log k$ and is motivated by the expected strength of the relevant $L$-value estimates. The conjecture is not resolved in the supplied text.