  Serre's uniformity question asks whether there exists a bound $N>0$ such
that, for every non-CM elliptic curve $E$ over $\mathbb{Q}$ and every prime
$p>N$, the residual Galois representation
$\rho_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to
\operatorname{Aut}(E[p])$ is surjective.
  The work of many authors has shown that, for $p>37$, this representation is
either surjective or has image contained in the normaliser of a non-split
Cartan subgroup $C_{ns}^+(p)$. Zywina has further proved that, whenever
$\rho_{E,p}$ is not surjective for $p>37$, its image can be either
$C_{ns}^+(p)$ or an index-3 subgroup of it. Recently, Le Fourn and Lemos showed
that the index-3 case cannot arise for $p>1.4 \cdot 10^7$. We show that the
same statement holds for any prime larger than $37$.
