On a conjecture on shifted primes with large prime factors

Abstract: Let $\mathcal{P}$ be the set of all primes and $\pi(x)$ be the number of primes up to $x$. For any $n\ge 2$, let $P^+(n)$ be the largest prime factor of $n$. For $0<c<1$, let $$T_c(x)=#{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c}.$$ In this note, we proved that there exists some $c<1$ such that $$\limsup_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}<\frac12,$$ which disproves a conjecture of Chen and Chen.