The exponent of harmonic LCM avoidance

Fix $k\ge 3$, and let $f_k(N)$ be the largest harmonic sum of a subset of $[N]$ containing no $k$ distinct integers with a common pairwise least common multiple. We prove that $f_k(N)=(\log N)^{γ_k+o(1)}$ for a well-defined exponent $γ_k\in(0,1]$. Following the weighted-pressure idea of Chojecki, we