Maximal nondegeneracy conjecture for the two-variable $p$-adic height

Let $r=\operatorname{rk}E(K)$, let $r^+$ and $r^-$ be the ranks of the $+1$ and $-1$ eigenspaces of complex conjugation on $E(K)$, and let $R_p(E,K)^{r-j,j}$ denote the component of the two-variable $p$-adic regulator in the summand with $r-j$ cyclotomic factors and $j$ anticyclotomic factors.

Maximal nondegeneracy conjecture. If $j$ is even and $0\leq j\leq 2\min(r^+,r^-)$, then

$$R_p(E,K)^{r-j,j}\ne 0.$$

The supplied status evidence says this statement is proved using the kernel property of anticyclotomic universal norms for the anticyclotomic $p$-adic height pairing.