The Eisenstein-space conjecture for prime-detecting quasimodular forms

Let $\mathcal{E}$ be the quasimodular Eisenstein space, generated additively by the even-weight Eisenstein series and their derivatives. Let $\Omega$ be the space of prime-detecting quasimodular forms: a quasimodular form $f=\sum_{n\geq0}b_n(f)q^n$ belongs to $\Omega$ when $b_n(f)\geq0$ for positive $n$, and $b_n(f)=0$ for $n\geq2$ if and only if $n$ is prime. The Eisenstein-space conjecture. We have

$$\Omega=\mathcal{E}\cap\Omega.$$

Equivalently, every prime-detecting quasimodular form should lie in the quasimodular Eisenstein space. The claim is based on numerical evidence, and the supplied text gives no proof or resolution.