Last's absolutely continuous spectrum conjecture for decaying Jacobi matrices

Let $J$ be the discrete Schrödinger operator on the half-lattice with bounded real potential $(v_n)$, represented by the Jacobi matrix with diagonal entries $v_n$ and off-diagonal entries $1$. Write $\sigma_{ac}(J)$ for its absolutely continuous spectrum. Last's conjecture. If

$$v_n\to 0$$

and, for a fixed $q\in\mathbb{Z}^+$,

$$v_{n+q}-v_n\in\ell^2,$$

then

$$\sigma_{ac}(J)=[-2,2].$$

The conjecture concerns the persistence of the absolutely continuous spectrum under a decaying potential whose fixed-step differences are square-summable. The paper's abstract states that the conjecture is proved, so its resolution is supplied by the source context.