Erdős Problem #1201 — Large Prime Divisors in Products of Consecutive Integers

For $\epsilon,\eta>0$, let $P\bigl(n(n+1)\cdots(n+k)\bigr)$ denote the greatest prime divisor of the product. Is it true that there exists $k\in\mathbb N$ such that the lower asymptotic density of the set

$$\left\{n\in\mathbb N:P\bigl(n(n+1)\cdots(n+k)\bigr)>n^{1-\epsilon}\right\}$$

is at least $1-\eta$? In precise limit form, does there exist $k$ for every $\epsilon>0$ and $\eta>0$ such that

$$\liminf_{x\to\infty}\frac{\#\{n<x:P(n(n+1)\cdots(n+k))>n^{1-\epsilon}\}}{x}\ge 1-\eta?$$