  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.
