The Fourier-coefficient growth criterion for cuspidality of Siegel modular forms

Let $c\in\mathbf R$, let $k_0\in\mathbf N$ be a scalar weight, and let $F\in M_k^n(\Gamma)$, where $\Gamma\subset\operatorname{Sp}_n(\mathbf Z)$ is a congruence subgroup. For all $k\geq k_0$, suppose that at some cusp there is a bound

$$a_F(T)\ll_F\det(T)^c$$

for every $T\in\Lambda_n^+$ and for some $c<k-\frac{n+1}{2}$. The Fourier-coefficient growth conjecture. Then $F\in S_k^n(\Gamma)$. This conjecture asks when growth of Fourier coefficients characterizes cuspidality; the paper presents results proving the criterion in specified settings, while the stated general form is not assigned a resolution here.