Bertolini–Darmon–Mazur conjecture on derived Selmer dimensions

Let $F$ be a quadratic imaginary field, let $F_\infty$ be its anticyclotomic $\mathbf Z_p$-extension, and let $A=E\times_{\mathbf Q}F$ for an elliptic curve $E/\mathbf Q$ satisfying the Heegner hypothesis that all primes of bad reduction split in $F$. Define

$$s^+=\dim_{\mathbf Q_p}S_p^{(1)}(A/F)^+,\qquad s^-=\dim_{\mathbf Q_p}S_p^{(1)}(A/F)^-.$$

Bertolini–Darmon–Mazur conjecture. In this situation,

$$\dim_{\mathbf Q_p}S_p^{(2)}(A/F)=|s^+-s^-|,\qquad \dim_{\mathbf Q_p}S_p^{(3)}(A/F)=1.$$

In the stated anticyclotomic setting, it is known that $e_\infty=1$, so the first derived height pairing is degenerate; the conjecture predicts the precise dimensions of the next derived Selmer groups, beyond the lower bound arising from complex conjugation.