Simon's absolutely continuous spectrum conjecture for multidimensional Schrödinger operators

Let $H=-\Delta+V(x)$ act on $\mathbb{R}^{d}$, where $V$ is a potential satisfying

$$\int\limits_{\mathbb{R}^{d}}\frac{V^{2}(x)}{|x|^{d-1}+1}\,dx<+\infty.$$

Here $\sigma_{ac}(H)$ denotes the absolutely continuous spectrum of $H$, and $\mathbb{R}^{+}$ denotes the nonnegative real axis. Simon's conjecture. Under these assumptions,

$$\sigma_{ac}(H)=\mathbb{R}^{+}.$$

The conjecture concerns preservation of absolutely continuous spectrum under slowly decaying perturbations of the free Schrödinger operator in dimensions greater than one. The source states that, despite some progress, it remains open.