Erdős problem 1201

Is it true that for every $\epsilon,\eta>0$ there exists a $k$ such that the density of $n$ for which $P(n(n+1)\cdots(n+k))>n^{1-\epsilon}$ is at least $1-\eta$, where $P(m)$ is the greatest prime divisor of $m$? A short argument via the Matomäki-Radziwiłł theorem establishes the lower-density version.