Near-quadratic growth conjecture for similar copies in general position

Let $m\geq3$ be a positive integer and let $P$ be a finite pattern with no $m$ collinear points. For every real $\varepsilon>0$, there is $N(\varepsilon)>0$ such that for all $n\geq N(\varepsilon)$,

$$S_{P}(n,m)\geq n^{2-\varepsilon}.$$

Near-quadratic growth conjecture. The bound above should hold for every such $m$ and $P$. A proof cannot follow from the paper's main theorem and would require a different construction of sets in general position with many similar copies of $P$; the claim is presented as an expectation rather than an established result.