Brent's conjecture on polynomial interpolation of normalized Hecke-group invariants

Let $m\geq 3$ and write

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

Brent's interpolation conjecture. For every integer $n>-2$, there exists a polynomial $A_n(x)\in\mathbb{Q}[x]$ such that $\alpha_n(m)=A_n(m)$ for $m=3,4,\ldots$, with $A_{-1}(x)\equiv 1$, $A_0(x)=24(x^2+4/3)$, and $A_1(x)=276x^2(x^4-8x^2/69-16/23)$. For $n>1$, the conjecture further asserts

$$A_n(x)=\overline a_n(x-2)(x+2)x^{n+1}a_n(x),$$

where $\{\overline a_n\}$ is the McKay–Thompson series of class $4A$, $a_n(x)$ is monic and irreducible over $\mathbb{Q}$ of degree $2n$, and the coefficient of $x^k$ in $a_n$ is zero if and only if $k$ is odd; it also includes the stated congruence divisibility and Galois-group assertions for prime $n$ and for $\operatorname{Gal}(a_n,\mathbb{Q})$. The claim predicts a structured polynomial family whose factorizations encode McKay–Thompson coefficients, congruences, and large Galois groups; the paper reports numerical evidence but does not establish these assertions.