Conjecture on the lower density of $\mathcal{W}_2$

Let $\mathcal{W}_2$ be the set of all positive integers $a$ for which there is no integer $m>1$ such that $(a-2^k,m)>1$ for every positive integer $k$. The $\mathcal{W}_2$ density conjecture. The set $\mathcal{W}_2$ has a positive lower asymptotic density. The paper poses this for further research and does not report a resolution.