Yui–Zagier's factorization conjecture for Weber CM values

Let $E_i=\mathbb{Q}(\sqrt{d_i})$ be imaginary quadratic fields of fundamental discriminants $d_i$ with $(d_1,d_2)=1$, let $F=\mathbb{Q}(\sqrt D)$ with $D=d_1d_2$, and let $E=\mathbb{Q}(\sqrt{d_1},\sqrt{d_2})$. Let $\tau_{\mathfrak a_i}$ be the CM points associated to integral ideal representatives $\mathfrak a_i$ of ideal classes in $\operatorname{Cl}(E_i)$, chosen with norm prime to $2$, and let $\rho(\mathfrak a)$ count ideals of $E$ with relative norm $\mathfrak a$. For $d_1\equiv d_2\equiv1\pmod 8$, define

$$\omega_2(\tau)=2^{12}q\prod_{n>0}(1+q^n)^{24}=2^{12}\frac{\Delta(2\tau)}{\Delta(\tau)}.$$

For $t=(m+\sqrt D)/2$ satisfying $|m|<\sqrt D$, $m$ odd, and $m^2\equiv D\pmod {16}$, let $\mathfrak p_t$ be the unique prime ideal of $F$ above $2$ such that $\operatorname{ord}_{\mathfrak p_t}(t\mathcal O_F)\geq1$. Yui–Zagier's factorization conjecture. One has

$$\sum_{[\mathfrak a_i]\in\operatorname{Cl}(E_i)}\log\left|\omega_2(\tau_{\mathfrak a_1})-\omega_2(\tau_{\mathfrak a_2})\right|^2 =\sum_{\substack{t=(m+\sqrt D)/2\\ |m|<\sqrt D,\ m\text{ odd}\\ m^2\equiv D\pmod {16}}}\ \sum_{\mathfrak p\text{ inert in }E/F}\frac{1+\operatorname{ord}_{\mathfrak p}(t\mathcal O_F)}2\,\rho(t\mathfrak p^{-1}\mathfrak p_t^{-2})\log\operatorname{N}(\mathfrak p).$$

This is the $a=24$ case of the conjectural factorization proposed by Yui and Zagier for differences of CM values. The paper discusses a potential obstruction: the big CM cycle used in the Bruinier–Kudla–Yang framework may generally be larger than the ideal class groups appearing in the proposed formula, so the conjecture's validity remains unclear.