Brent's nonvanishing conjecture for Fourier coefficients of Hecke-group invariants

For $m\geq 3$, write

$$j_m(z)=\frac{1}{q_m(z)}+\sum_{n\geq 0}\alpha_n(m)q_m(z)^n.$$

Brent's nonvanishing conjecture. For each fixed integer $n\geq -1$ and every integer $m\geq 3$, one has

$$\alpha_n(m)\neq 0.$$

This follows formally from the preceding interpolation conjecture if the conjectured polynomial factorization has no roots among integers $m\geq 3$; the paper notes that positivity at $m=3$ is known, but the general assertion remains conjectural.