The long-arithmetic-progression conjecture for digit positions

For $\beta>1$, let $A_k(\beta)$ denote the set of positions of digit $1$ in the binary expansion of $\beta^k$. Long-arithmetic-progression conjecture. If $\log\beta/\log2$ is irrational, then

$$W=\{k\in\mathbb{N}:A_k(\beta)\text{ does not have arbitrarily long arithmetic progressions}\}$$

has natural density $0$. This extends the paper's finite-progression results to arithmetic progressions of unbounded length; no proof or disproof is supplied.