The logarithmic exceptional-set conjecture for powers of 3

Let $A_k$ be the set of positions of digit $1$ in the binary expansion of $3^k$, and let

$$W=\{k\in\mathbb{N}:A_k\text{ does not have a 3-term arithmetic progression}\}.$$

Logarithmic exceptional-set conjecture. Under these conditions,

$$\#(W\cap[1,N])=O(\log N).$$

The paper has already proved the weaker estimate $O_\epsilon(N^\epsilon)$ for every $\epsilon>0$; the logarithmic bound is posed as a further open problem.