Hildebrand's consecutive stable-set intersection conjecture

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 $0$. Write $\underline{d}(A)$ for the lower density of $A$.

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

$$\underline{d}\bigl(\mathcal{S}\cap(\mathcal{S}+1)\cap(\mathcal{S}+2)\cap\cdots\cap(\mathcal{S}+(k-1))\bigr)>0.$$

This conjecture generalizes the known two-set result that stable sets of positive lower density contain many consecutive pairs. The paper provides a counterexample, so the conjecture is refuted.