Erdős Problem #37 — Lacunary sequences that fail to be essential components

A sequence A is called by Khintchin an essential component if for every sequence B with $d_s(\mathrm{B}) > 0$, $d_s(\mathrm{A} + \mathrm{B}) > d_s(\mathrm{B})$. [...] This led me to the following conjecture: Let $n_1 < n_2 < \ldots$ be an infinite sequence of integers satisfying $\frac{n_{k-1}}{n_k} > c > 1$ [sic in the original; presumably $\frac{n_{k+1}}{n_k}$ is meant], then our sequence can not be an essential component. I was so far unable to prove this conjecture.