Nonvanishing conjecture for the first Fourier coefficient of twisted divisor-function forms

Let $D$, $cell$, $k$, and $e$ be the parameters defining the modular form $\mathcal{F}_{D,\ell,k,e}$ and its Fourier coefficient $a_{D,\ell,k,e}(1;\chi)$, and set

$$K=k+\ell+2e.$$

Nonvanishing conjecture. The first Fourier coefficient satisfies

$$a_{D,\ell,k,e}(1;\chi)\neq 0$$

for $K\gg_D1$.

This conjecture proposes nonvanishing of the first Fourier coefficient when the weight is sufficiently large relative to $D$. The source notes that a related result is known when $e=0$ and $k=\ell$, after modifying the definition, under the explicit bound $K\geq 10D+2$; the general assertion remains open.