The asymptotic consecutive-pattern distribution conjecture

For $k\geq 3$, let $\pi\in S_k$ be a consecutive pattern, let $I\subseteq\mathbb{Z}^+$ be finite, let $\mathcal{P}_{\pi}(I,n)$ be the set of permutations in $S_n$ whose consecutive-pattern occurrence starting indices are exactly $I$, and write

$$p_{\pi}(I,n)=|\mathcal{P}_{\pi}(I,n)|,\qquad p_{\pi}(n)=p_{\pi}(\emptyset,n).$$

The asymptotic consecutive-pattern distribution conjecture. For any $k\geq 3$, any $\pi\in S_k$, and any finite $I\subseteq\mathbb{Z}^+$, there are constants $c_{\pi,I},\alpha\in\mathbb{R}$ with $\alpha>0$ such that

$$p_{\pi}(I,n)=c_{\pi,I}p_{\pi}(n)\left(1+O\left(n^{-\alpha}\right)\right).$$

This conjectures that prescribing any finite set of consecutive-pattern occurrences changes the unrestricted count only by an asymptotically constant factor, with a power-saving error term. The source presents it as an analogue of an earlier theorem for decreasing patterns and states that the question remains open.