Full box-dimension conjecture for alpha-beta sequences

Let $\alpha$ and $\beta$ be real numbers such that $1,\alpha,\beta$ are independent over the field of rational numbers. An $\alpha\beta$-sequence is a sequence $\{x_n\}_{n\geq 1}$ with $x_1=0$ such that, for each $i\geq 1$, one may choose freely between $x_{i+1}=x_i+\alpha\mod 1$ and $x_{i+1}=x_i+\beta\mod 1$. Full box-dimension conjecture. Every $\alpha\beta$-sequence has full box dimension. The supplied text gives no resolution of this problem; its significance is that it asks whether the rational independence of the two increments forces every permitted deterministic path to have maximal box dimension.