Serre's uniformity question for elliptic-curve Galois representations

Let $E$ be a non-CM elliptic curve over $\mathbb{Q}$, let $\ell$ be a prime, and let $\rho_{E,\ell}$ denote the mod $\ell$ Galois representation.

Serre's uniformity question. There exists a bound $C$ such that for every non-CM elliptic curve $E/\mathbb{Q}$ and every prime $\ell>C$, the representation $\rho_{E,\ell}$ is surjective.

This is a uniform strengthening of the open-image theorem for non-CM elliptic curves, asserting that all sufficiently large prime-level representations are surjective with one bound independent of $E$. The source presents it as a problem originally posed by Serre and uses it as an assumption for finiteness results.