Near-linear upper-bound conjecture for parallelogram-free patterns

Let $P$ be a parallelogram-free pattern. For every real $\varepsilon>0$, there is $N(\varepsilon)>0$ such that for all $n\geq N(\varepsilon)$,

$$S_{P}^{\nparallel}(n)\leq n^{1+\varepsilon}.$$

Near-linear upper-bound conjecture. The function $S_{P}^{\nparallel}(n)$ should satisfy this bound for every parallelogram-free pattern $P$. The paper suggests that the construction giving the existing lower-bound behavior is close to optimal, but that a stronger upper-bound argument is needed; the conjecture remains open.