Character-occurrence finiteness conjecture for elliptic curves over the maximal abelian extension

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $X$ be the set of even characters of $\operatorname{Gal}(\mathbb{Q}^{\operatorname{ab}}/\mathbb{Q})$. Character-occurrence finiteness conjecture. The set

$$\{\chi\in X:\ \operatorname{order}(\chi)\geq 7,\ \operatorname{order}(\chi)\neq 8,10\text{ or }12,\text{ and }\chi\text{ occurs in }E(\mathbb{Q}^{\operatorname{ab}})\}$$

is finite. The source presents this as a conjectural restriction on character components of Mordell–Weil groups over the maximal abelian extension, with no resolution supplied.