On a conjecture on shifted primes with large prime factors

Let P be the set of all primes and pi(x) be the number of primes up to x. For any n >= 2, let P+(n) be the largest prime factor of n. For 0 < c < 1, let T-c(x) = # {p <= x : p is an element of P , P+(p - 1) >= p(C)}. In this note, we prove that there exists some c < 1 such that lim sup (x ->infinity) T-c(x)/pi(x) < 1/2, which disproves a conjecture of Chen and Chen.