Quantitative lower bound for Pythagorean pairs in boxes of prime powers

For each positive integer $N$, let $p_i$ be the $i$-th prime for $i\in \{1,\dots ,N\}$, and set $${\mathrm{\Phi }}_N=\{\prod _{i=1}^N{p}_i^{a_i}:a_i\in \{0,1,\dots ,N-1\}\}\mathrm{.}$$ Write $f(N)$ for the largest cardinality of a subset of ${\mathrm{\Phi }}_N$ containing no two distinct legs of an integral right triangle. We prove $$N^{N-1}(\log \log \log N)\ll f(N)\mathrm{.}$$