Chen–Chen conjecture on large prime factors of shifted primes

Let $\mathcal{P}$ be the set of all primes, let $\pi(x)$ denote the number of primes up to $x$, and, for an integer $n\geq 1$, let $P^+(n)$ denote its largest prime factor, with $P^+(1)=1$. For $0<c<1$, define

$$T_c(x)=\#\{p\le x:p\in\mathcal{P},P^+(p-1)\ge p^c\}.$$

Chen–Chen conjecture. For any integer $k\ge 1$ and any $c\in [\frac{1}{k+1},\frac{1}{k})$, we have

$$T_c(x)\ge \left(1-\frac{1}{k+1}\right)\frac{x}{\log x}+o\left(\frac{x}{\log x}\right).$$

The conjecture extends lower bounds for the number of shifted primes whose predecessor has a large prime factor. The paper proves that it is false in the broad sense suggested by the conjecture: for some $c<1$, the proportion represented by $T_c(x)$ has limsup strictly below $1/2$, thereby disproving the conjecture of Chen and Chen.