Closure-interval conjecture for powers of two and irrational rotations

Let $\alpha$ be an irrational number and $d$ be a real number. Consider the sequence $\{2^n d+n\alpha\mod 1\}_{n\geq 1}$ in $[0,1]$. Closure-interval conjecture. The topological closure of this sequence contains intervals. The conjecture asks whether the deterministic sequence formed by combining an irrational rotation with the powers-of-two sequence always has a closure containing a nontrivial interval; the supplied text gives no resolution.