This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM elliptic curve $E$ defined over a number field $K$:

For each prime $\ell$, the index $I(E,\ell)$ of the image of $\rho_{E,\ell}$ in $\operatorname{GL}_2(\mathbb{Z}_\ell)$ is finite (and bounded independent of $\ell$), and

The representation $\rho_{E,\ell}$ is surjective for all but finitely many primes $\ell$.

Remark: As I understand it, these two statements together say that the representation $\rho_E:G_K \to \operatorname{GL}_2(\hat{\mathbb{Z}})$ made up of all the $\ell$-adic ones has open image, and hence "Serre's Open Image Theorem" refers to both of these statements, although many papers written on this topic seem to only discuss one statement or the other.

Let $p_0(E)$ denote the smallest prime such that $\rho_{E,\ell}$ is surjective for all $\ell > p_0$.

**My question: What are the strongest conjectured ("conjectured" means either officially conjectured or in a folklore way, or even someone writing down that they hope it's true) generalizations of Serre's Theorem which are uniform in $E$?**

Here are examples of the kind of statements I am referring to:

Conjecture 1: $p_0(E)$ is bounded independently of $E$, where $E$ ranges over all elliptic curves defined over $\mathbb{Q}$ (and the bound is 37).

Conjecture 2: $I(E,\ell)$ is bounded independently of $E$ and $\ell$, where $E$ ranges over all elliptic curves defined over $\mathbb{Q}$ (and the bound is...?).

Conjecture 1': Like Conjecture 1, but for elliptic curves defined over an arbitrary number field $K$, and the bound only depends on $K$.

Conjecture 2': Like Conjecture 2, but for elliptic curves defined over an arbitrary number field $K$, and the bound only depends on $K$.

Conjecture 1'': Like Conjecture 1', but the bound only depends on $[K:\mathbb{Q}]$.

Conjecture 2'': Like Conjecture 2', but the bound only depends on $[K:\mathbb{Q}]$.

I am aware that a lot of progress has been made on Conjecture 1, and we might be close to knowing that one.