The X(n)–telescope meet conjecture

Let $X(n)$ be the standard spectra in the Ravenel sequence and let $T(k)$ be the telescope of a $v_k$-self map. Their Bousfield classes are denoted by $\langle X(n)\rangle$ and $\langle T(k)\rangle$.

X(n)–telescope meet conjecture. For all $n$ and $k$,

$$\langle X(n)\rangle\wedge\langle T(k)\rangle=\langle T(k)\rangle.$$

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