Asymptotic $L^2$-mass conjecture for pullbacks of Saito–Kurokawa lifts

Let $N$ tend to infinity through odd square-free integers. For an SK lift $F_f\in \mathrm{SK}_{k+1}^{new}(N)$, let $F_f^\circ$ be its pullback and let $N(F_f)$ be its normalized $L^2$-mass. Write $\phi$ for Euler's totient function, $\zeta_{(d)}$ for the zeta function with Euler factors at primes dividing $d$ removed, and $w_f(p)$ for the Atkin–Lehner eigenvalue of $f$ at $p$. Asymptotic $L^2$-mass conjecture. There exists some $\delta>0$ such that

$$N(F_f)= \frac{2\phi(N)\zeta_{(N)}(2)}{N}\prod_{p\mid N}\left(1+\frac{w_f(p)}{p}\right)^{-1}\sum_{d\mid N}\frac{\zeta_{(d)}(2)}{\phi(d)} \prod_{p\mid d}\left(1+\frac{1}{p}\right)^3(1+w_f(p))^2 + O(N^{-\delta}).$$

When $N=p$ is prime, this becomes

$$N(F_f)=2+O(p^{-\delta}).$$

The conjecture predicts a precise limiting size for the normalized pullback mass, refining the preceding non-vanishing question. It is proposed as an asymptotic statement as the odd square-free level grows; the source does not provide evidence of a resolution.