Craig–van Ittersum–Ono conjecture on prime-detecting quasi-modular forms

Let $q=e^{2\pi i\tau}$, and let $\Omega$ be the subset of the graded ring of integer-weight quasi-modular forms of full level consisting of forms

$$f(\tau)=\sum_{n\geqslant 0}c_f(n)q^n$$

whose Fourier coefficients $c_f(n)$ strongly detect the primes. Let $\mathcal{E}$ be the vector space of quasi-modular Eisenstein series, spanned by Eisenstein series and their derivatives. Craig–van Ittersum–Ono conjecture. With this notation,

$$\Omega\subset\mathcal{E}.$$

The conjecture asserts that every quasi-modular form whose Fourier coefficients strongly detect the primes is a quasi-modular Eisenstein series. The source provides no resolution, so the conjecture remains open.