BSD regulator-quotient square conjecture for elliptic curves

Let $E/K$ be an elliptic curve over a number field $K$, with a fixed invariant differential $\omega$. Let $L_i/K$ and $L'_j/K$ be finite extensions such that the associated permutation representations satisfy

$$\bigoplus_i \operatorname{Ind}_{L_i/K} \mathbf{1}_{L_i}\simeq \bigoplus_j \operatorname{Ind}_{L'_j/K} \mathbf{1}_{L'_j}.$$

Write $\operatorname{Reg}(E/L)$ for the regulator and $C(E/L)$ for the product of local Tamagawa and differential factors.

BSD regulator-quotient square conjecture. Then

$$\prod_i \operatorname{Reg}(E/L_i)C(E/L_i)\equiv \prod_j \operatorname{Reg}(E/L'_j)C(E/L'_j)\pmod{\mathbb{Q}^{*2}}.$$

This removes Tate–Shafarevich and torsion contributions by working modulo rational squares. It is presented as a consequence of the Birch–Swinnerton-Dyer conjecture and is open to the extent that the required BSD input is open.