The no-large-prime coincidence conjecture for elliptic curves

Let $p$ be a prime, and let a $p$-coincidence be an equality of division fields

$$\mathbb{Q}(\mathcal{E}[j])=\mathbb{Q}(\mathcal{E}[pj]).$$

The no-large-prime coincidence conjecture. If $p\geq 5$, then there are no non-CM elliptic curves $\mathcal{E}$ over $\mathbb{Q}$ with a $p$-coincidence. This is a more focused formulation of the claim that only the primes $2$ and $3$ can occur; it remains open in the supplied text.