On the largest prime factors of consecutive square-free integers

For an integer $n \gt 1$, let $P^+(n)$ be the largest prime factor of $n$. Following a celebrated conjecture of Erdős and Turán in the 1930s, Erdős and Pomerance proved in 1978 that $$\liminf _{x\rightarrow \infty }\frac{|\{n\le x:P^+(n+1) \gt P^+(n)\}|}{x} \gt 0. $$ In this article, their result is extended to $$ \liminf_{x\rightarrow \infty}\frac{|\{n\le x:P^+(n+1) \gt P^+(n),\, \mu ^2(n)=\mu ^2(n+1)=1\}|}{x} \gt 0. $$