On a conjecture on shifted primes with large prime factors, II

Abstract: Let $\mathcal{P}$ be the set of primes and $\pi(x)$ the number of primes not exceeding $x$. Let also $P^+(n)$ be the largest prime factor of $n$ with convention $P^+(1)=1$ and $$ T_c(x)=#\left{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\right}. $$ Motivated by a 2017 conjecture of Chen and Chen, we show that for any $8/9\le c<1$ $$ \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\le 8(1/c-1), $$ which clearly means that $$ \limsup_{x\rightarrow\infty}T_c(x)/\pi(x)\rightarrow 0, \quad \text{as}~c\rightarrow1. $$