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

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 $$T_c(x)=\mathrm{\#}\{p\le x:p\in \mathcal{P},P^{+}(p-1)\ge p^c\}\mathrm{.}$$ Motivated by a 2017 conjecture of Chen and Chen, we show that for any $8\mathrm{/}9\le c<1$ $$\underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}{T}_c(x)\mathrm{/}\pi (x)\le 8(1\mathrm{/}c-1),$$ which clearly means that $$\underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}{T}_c(x)\mathrm{/}\pi (x)\to 0,\text{as }c\to 1.$$