Klick–Strungaru–Tcaciuc conjecture on arithmetic progressions in cut-and-project sets

Let $G$ be a locally compact second countable group and let $\Lambda \subset G$ be a cut-and-project set. A subset of $G$ is an arithmetic progression of length $r$ if it has the form $\{\lambda_o+k\lambda:1\leq k\leq r\}$ for some $\lambda_o,\lambda\in G$ with $\lambda\neq 0$. A subset has positive upper Banach density when its upper Banach density in $G$ is positive. Klick–Strungaru–Tcaciuc conjecture. Every subset $P_o \subset \Lambda$ with positive upper Banach density contains arithmetic progressions of arbitrarily large finite length. The paper states that its main theorem provides a solution to this conjecture, so the claim is presented here as resolved.