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

Let P be the set of primes and π(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 Tc(x)=#≤ x:p∈ P,P⁺(p-1)≥ pᶜ\. Motivated by a 2017 conjecture of Chen and Chen, the author [6] proved that there exists some absolute constant $c<1$ such that {align}{eq1} x→∞Tc(x)/π(x)<1/2. {align} In this note, the prior result is considerably improved to x→∞Tc(x)/π(x)→ 0, as~c→1.