Asymptotic order of near-equal consecutive largest prime factors

Determine whether the number of integers \(n\le x\) satisfying \(1\le P^+(n+1)/P^+(n)<1+\varepsilon\) has order \(x/\log x\) for each fixed \(\varepsilon>0\), specifically proving an asymptotic-order relation \(\asymp_\varepsilon x(\log x)^{-1}\).

Background

The paper proves a lower bound of order $x/\log x$ for ratios of consecutive largest prime factors lying in a fixed one-sided interval near 1. The authors then state a heuristic conjecture that this lower-bound scale is the correct order of magnitude.