De Koninck–Doyon permutation-pattern conjecture

Establish, for every fixed integer \(k\ge 2\) and every permutation \((a_1,\ldots,a_k)\) of \((0,1,\ldots,k-1)\), that \(\#\{n\le x:P^+(n+a_1)<\cdots<P^+(n+a_k)\}\sim x/k!\) as \(x\to\infty\).

Background

The conjecture generalizes comparisons of largest prime factors from two or three consecutive integers to arbitrary fixed blocks. It predicts that all strict orderings of the largest prime factors occur with equal asymptotic frequency, despite the arithmetic dependence among the shifted integers.