The equidistribution conjecture for separated consecutive-pattern occurrences

Fix $k\geq 3$, let $\pi\in S_k$, and let $a\in\mathbb{Z}^+$. For $n\in\mathbb{Z}^+$ and $I=\{i_1,i_2,\ldots,i_a\}\subseteq[n]$, suppose that

$$i_1\geq\sqrt n,\qquad i_{j+1}-i_j\geq\sqrt n\quad(1\leq j\leq a-1),\qquad n-i_a\geq\sqrt n.$$

Here $p_{\pi}(I,n)$ counts permutations in $S_n$ with consecutive-pattern occurrence set $I$, and $p_{\pi}(n)=p_{\pi}(\emptyset,n)$. The equidistribution conjecture for separated consecutive-pattern occurrences. There is a real constant $C_{\pi,a}>0$ such that

$$p_{\pi}(I,n)=C_{k,a}p_{\pi}(n)\left(1+O_{\pi,a}\left(n^{-\alpha}\right)\right).$$

The claim is stated as an analogue of the source's equidistribution theorem for decreasing consecutive patterns. It asserts uniform asymptotic independence for occurrences separated from one another and from the endpoints by at least $\sqrt n$; the source gives no resolution status.