Irreducibility conjecture for the class polynomial of $\chi^*$

Let $\chi^*$ be the level $1$ almost holomorphic modular function defined by

$$\chi^*=1728\cdot\dfrac{E_2^*E_4E_6}{E_4^3-E_6^2},$$

where $E_4$ and $E_6$ are the usual Eisenstein series and $E_2^*(\tau)=E_2(\tau)-\dfrac{3}{\pi\operatorname{Im}\tau}$. For a discriminant $\delta$, let $P^1_\delta$ denote the relevant set of level $1$ positive definite quadratic forms, and let $\tau_Q$ be the associated quadratic point. Define the class polynomial

$$H^{\chi^*}_\delta=\prod_{Q\in P^1_\delta}\left(X-\chi^*(\tau_Q)\right).$$

Irreducibility conjecture for the class polynomial of $\chi^*$. The polynomial $H^{\chi^*}_\delta$ is irreducible over $\mathbb{Q}$, and hence

$$\mathbb{Q}(j(\tau))=\mathbb{Q}(\chi^*(\tau)).$$

Masser proved that $\chi^*(\tau)$ is algebraic for quadratic $\tau$ and that $\mathbb{Q}(\chi^*(\tau))\subseteq\mathbb{Q}(j(\tau))$; the conjecture asserts equality of these fields via irreducibility of the class polynomial.