Complete positivity conjecture for the quasimodular forms $Y_{w,2}$

Let $X_{w,2}(z)$ be the depth-two extremal quasimodular form of weight $w$, and define

$$Y_{w,2}(z) = X_{w,2}(z) - 2^{w-2} X_{w,2}(2z).$$

A quasimodular form is completely positive when all coefficients in the relevant complete-positivity sense are positive. Complete positivity conjecture. For all $w \ne 6$, $Y_{w,2}(z) = X_{w,2}(z) - 2^{w-2} X_{w,2}(2z)$ is completely positive. The paper establishes complete positivity in some cases, including forms used in its applications, but the assertion for every $w \ne 6$ remains open.