Disproved. Konieczny [Ko16b] proves that the set is not an additive basis of order two for almost every $\alpha>0$, and explicitly for $\alpha=\sqrt2$, with any $\epsilon(n)\to0$ in place of $1/\log n$; for thresholds decaying slowly enough (an $\alpha$-dependent rate that the paper does not relate to $1/\log n$) the set is a basis of order two for uncountably many exceptional $\alpha$ and of order three for every irrational $\alpha$. The accepted claim is Konieczny.