The sixth-power norm conjecture for differences of lambda invariants at CM points

Let $d_1$ and $d_2$ be the discriminants under consideration, with $d_1 e d_2$ and $d_1 e 0$, $d_2 e 0$. Assume $d_1 equiv d_2 equiv 5\pmod 8$. For each $t=\frac{m+\sqrt D}{2}\in\mathcal O_F$, let $\mathfrak p_t$ be the unique \prime ideal of $F$ above $2$ with $2\in\mathfrak p_t$. Then

$$\sum_{\substack{t=\frac{m+\sqrt D}{2}\in\mathcal O_F,\,\operatorname{ord}_{2}(m^2-D)\text{ odd}\\ |m|<\sqrt D,\,m\equiv -1\pmod 4}}\rho_{E/F}(t\mathfrak p_t^{-1})$$

is a multiple of $3$. The sixth-power norm conjecture. The norm $\operatorname{N}\!\left(\lambda\!\left(\frac{d_1+\sqrt{d_1}}{2}\right)-\lambda\!\left(\frac{d_2+\sqrt{d_2}}{2}\right)\right)$ is a sixth power up to sign. This conjectural divisibility is suggested by the tabulated norm values and the stated theorem, and would explain their sixth-power structure.