Kaneko–Koike positivity and denominator conjecture for extremal quasimodular forms

Let $\widetilde{f}_{\ell,w}$ be a normalized extremal quasimodular form of depth $\ell$ and weight $w$, with Fourier expansion

$$\widetilde{f}_{\ell,w}(z)=\sum_{n\geq 0}a_nq^n.$$

Kaneko–Koike's conjecture. If $\ell\leq 4$ and $w>2$, then every Fourier coefficient $a_n$ is positive. Moreover, no prime factor of the denominator of any such coefficient is greater than $w$.

This conjecture concerns the arithmetic and positivity properties of normalized extremal quasimodular forms. The supplied text reports it as a conjecture motivated by numerical experiments, without giving evidence of a resolution.