Depth‑2 recurrence analogous to the depth‑1 identity for extremal quasimodular forms

Develop a recurrence relation for depth‑2 extremal quasimodular forms X_{w,2}, analogous to the depth‑1 identity X_w' = (5w/72) X_6 X_{w-4} + (7w/72) X_8 X_{w-6} (equation (kkd1eq1)), that could lead to a complete proof of the Kaneko–Koike positivity conjecture in depth 2.

Background

For depth 1, the authors prove a new differential recurrence (equation (kkd1eq1)) expressing the derivative of the depth‑1 extremal form X_w in terms of lower‑weight extremal forms. This identity implies complete positivity for depth‑1 extremal forms.

In depth 2, they verify the positivity conjecture for small weights using exceptional identities but note that an analogue of the depth‑1 recurrence is lacking. Finding such a recurrence could provide a systematic route to proving positivity in depth 2.