Kaneko–Koike positivity conjecture for extremal quasimodular forms of depth at most 4

Prove that for every even weight w and depth s with 0 ≤ s ≤ 4 (subject to the standard constraints on extremal forms), the normalized extremal quasimodular form X_{w,s} ∈ QM_w^s(SL_2(Z)) has Fourier coefficients that are all strictly positive.

Background

Extremal quasimodular forms are depth‑s quasimodular forms of given weight w with maximal vanishing at the cusp. Existence and uniqueness (up to scaling) for depths s ≤ 4 have been established by Pellarin, and differential equations for these forms were developed by Grabner.

Kaneko and Koike conjectured that all Fourier coefficients of these extremal forms (for depths up to 4) are positive. The present paper proves the conjecture in depth 1 and provides additional evidence at small weights for depth 2, but the full conjecture for depths 2–4 remains unresolved.