Conjecture on positivity of coefficients of the quasimodular forms $f_w$

For even $w$, let $f_w$ be the quasimodular form defined by the paper's Propositions equivariant to weights $0$ and $2$, with Fourier expansion

$$f_w=\sum_{m=\left\lfloor\frac{w}{4}\right\rfloor-1}^{\infty}a_w(m)q^m.$$

Coefficient-positivity conjecture. Every coefficient in this expansion is positive:

$$a_w(m)>0\qquad\text{for }m\geq\left\lfloor\frac{w}{4}\right\rfloor-1.$$

The conjecture is motivated by numerical experiments and positivity has been checked for even $w$ from $8$ through $94$, corresponding to dimensions $4$ through $312$ divisible by $4$. It remains open in general.