Let $K$ be a number field (possibly of infinite degree over $\mathbb{Q}$) and $E$ an elliptic curve without complex multiplication.

Let $L:= K(E_{5^{\infty}7^{\infty}11^{\infty}...})$ be the field obtained by adjoining all torsion points of order prime to $6$ to $K$ and let $G:=\operatorname{Gal}(L/K)$.

Let $p\geq 5$ be a prime. Is the group $$(pE(L))^{G}/pE(K)$$ zero for almost all primes $p$?

I am particularly interested in the cases

(1) $K=\mathbb{Q}[\sqrt{-5},\sqrt{-7},\sqrt{-11},...]$

(2) $K=\mathbb{Q}$

EDIT: One must of course assume $\operatorname{rank}_{K}(E)\geq 1$.