The arithmetic-progression conjecture for digit ones in powers of 3

For each integer $k\geq 0$, write the binary expansion of $3^k$ as a sequence of digits in $\{0,1\}$, and let the positions of its digit $1$ be regarded as a subset of the nonnegative integers. Arithmetic-progression conjecture. For all but finitely many integers $k\geq 0$, this set of positions contains a $3$-term arithmetic progression. This is the motivating number-theoretic problem of the paper; the paper proves the weaker bound that the exceptional set up to $N$ is $O_\epsilon(N^\epsilon)$ for every $\epsilon>0$, while finiteness remains open.