Refinement of the Craig–van Ittersum–Ono conjecture for prime-detecting quasimodular forms

Let $N$ be a positive integer. Let $\beta\widetilde{E}(N)$ denote the space of quasimodular Eisenstein forms on $\Gamma_{0}(N)$, let $\Omega_{N}$ denote the space of prime-detecting quasimodular forms of level $N$, and let $\widetilde{S}^{\mathrm{old}}(N)$ denote the space of quasimodular oldforms on $\Gamma_{0}(N)$. Refined conjecture.

$$\Omega_{N}=\Omega_{N}\cap\left(\widetilde{E}(N)\oplus\widetilde{S}^{\mathrm{old}}(N)\right).$$

This refinement is motivated by examples whose nonzero cuspidal parts lie in the quasimodular oldform space. It predicts that any failure of the Eisenstein-space inclusion is accounted for by oldforms, while the claim remains open in the source.