The sphere and complex projective space Bousfield-equivalence conjecture

Let $S$ be the sphere spectrum and let $\mathbf{C}P^{\infty}$ be infinite complex projective space, regarded through its suspension spectrum.

Sphere–projective-space conjecture. Their Bousfield classes are equal:

$$\langle S\rangle=\langle\mathbf{C}P^{\infty}\rangle.$$

The source says that Hopkins had proved this statement, although the authors had not seen a proof. Thus it is solved.