Non-eigenform conjecture for the forms $\mathcal{F}_{D,k,e}$

Let $D$, $k$, and $e$ be parameters such that the cusp-form space $S_{2k+2e}$ has dimension greater than one, and let $\mathcal{F}_{D,k,e}$ denote the modular form constructed in the source.

Non-eigenform conjecture. If

$$\dim S_{2k+2e}>1,$$

then $\mathcal{F}_{D,k,e}$ is not a Hecke eigenform.

The question concerns the case $\ell=k$, where the source explains that nonvanishing of the relevant twisted central $L$-values is not known in general. A special case with $D=1$ and $e=0$, using a modified definition, has been studied previously; the stated general assertion remains open.