Mazur–Rubin's conjecture on Selmer near-companion elliptic curves

Let $K$ be a number field, let $p$ be a prime, and let $E_1$ and $E_2$ be elliptic curves over $K$. They are $p$-Selmer near-companions over $K$ if there exists a constant $C=C(E_1,E_2,K)$ such that for every $\chi\in\operatorname{Hom}(G_K,\{\pm1\})$,

$$\left|\dim_p\operatorname{Sel}_p(E_2^\chi/K)-\dim_p\operatorname{Sel}_p(E_1^\chi/K)\right|<C.$$

Mazur–Rubin's conjecture. If $E_1$ and $E_2$ are $p$-Selmer near-companions over $K$, then there exists a $G_K$-module isomorphism

$$E_1[p]\cong E_2[p].$$

The conjecture relates uniform agreement of the dimensions of the quadratic twists' $p$-Selmer groups to the associated mod-$p$ Galois representations. Yu proved the conjecture when $n=2$, while the general statement is presented here as the conjecture of Mazur and Rubin.