Hildebrand's nonemptiness conjecture for consecutive stable-set intersections

Let $\mathcal{S}\subseteq\mathbb{N}$ be a stable set, meaning that for every natural number $d$, the set of $n$ for which membership of $n$ and $dn$ in $\mathcal{S}$ differs has density $0$.

Hildebrand's nonemptiness conjecture. If $\mathcal{S}$ has positive lower density, then for every natural number $k$,

$$\mathcal{S}\cap(\mathcal{S}+1)\cap(\mathcal{S}+2)\cap\cdots\cap(\mathcal{S}+(k-1))\neq\varnothing.$$

The source presents this as a weaker version of Hildebrand's density conjecture and notes that it is equivalent to positivity of the upper density of the same intersection. Since the paper's counterexample targets the stronger positive-lower-density assertion, the status of this weaker formulation should be checked separately.