Asymptotic parity conjecture for Fourier coefficients of the Hauptmodul $j^*_{10}$

Let $A$ and $B$ be positive integers such that

$$A\equiv 0\pmod 8,\qquad B\equiv 3\pmod 8.$$

Write $\mathcal{J}^*_{10}(m)$ for the $m$-th Fourier coefficient of $j^*_{10}(\tau)$. Asymptotic parity conjecture. There are such integers $A$ and $B$ for which

$$\lim_{X\to +\infty}\frac{\#\left\{n\leq X:\mathcal{J}^*_{10}(An+B)\equiv 0\pmod 2\right\}}{X}=1.$$

The conjecture proposes that, in a suitable progression with difference divisible by $8$ and offset congruent to $3$ modulo $8$, almost all of the corresponding Fourier coefficients are even. It is based on numerical calculations; the broader distribution of the Fourier coefficients of $j^*_{10}(\tau)$ remains to be studied.