Sato–Tate conjecture for non-CM elliptic curves

Let $E$ be an elliptic curve without complex multiplication. For each prime $p$, write the Frobenius trace as

$$a_p=2\sqrt{p}\cos\varphi_p.$$

Sato–Tate conjecture. The angles $\varphi_p$ are equidistributed in $[0,\pi]$ with respect to the Sato–Tate density measure

$$\frac{2}{\pi}\sin^2\varphi\,d\varphi.$$

The conjecture describes the distribution of Frobenius angles for non-CM elliptic curves. The source says that Sato computed the distribution and Tate gave theoretical evidence, but it does not state the later resolution of the conjecture.