De Koninck–Doyon conjecture on orderings of largest prime factors

Fix an arbitrary integer $k\geq 2$, and let $a_1,a_2,\ldots,a_k$ be any permutation of $0,1,\ldots,k-1$. Let $P^+(m)$ denote the largest prime factor of $m$. De Koninck–Doyon conjecture. As $x$ tends to infinity,

$$\#\{n\leq x:P^+(n+a_1)<P^+(n+a_2)<\cdots<P^+(n+a_k)\}\sim\frac{1}{k!}x.$$

This generalizes the Erdős–Turán ordering conjecture from two consecutive integers to every ordering of $k$ consecutive integers. The supplied text does not report a resolution, so the conjecture remains open here.