On Fourier coefficients for Siegel cusp forms of degree $n$

Let $F(Z)$ be a cusp form of integral weight $k$ relative to the Siegel modular group $Sp_n(\mathbb{Z})$ and let $f(N)$ be its Fourier coefficient with index $N$. Making use of Rankin's convolution, one proves the estimate
$$ f(N)=O\Bigl(|N|^{\frac k2-\frac17\delta(n)}\Bigr), \qquad (1) $$
where
$$ \delta(n)=\frac{n+1}{(n+1)\Bigl(zn+\frac{1+(-1)^n}{2}\Bigr)+1} $$
Previously, for $n\geq2$ one has known Raghavan's estimate
$$ f(N)=O(|N|^{\frac k2}) $$
In the case $n=2$, Kitaoka has obtained a result, sharper than (1), namely:
$$ f(N)=O\Bigl(|N|^{\frac k2-\frac14+\varepsilon}\Bigr) \qquad (2) $$
At the end of the paper one investigates specially the case $n=2$. It is shown that in some cases the result (2) can be improved to, apparently, unimprovable estimates if one assumes some analogues of the Petersson conjecture. These results lead to a conjecture regarding the optimal estimates of $f(N)$, $n=2$.