Asymptotic formula for the totient sum over prime quotients

Let $p$ run over primes, let $\phi$ denote Euler's totient function, and let $[y]$ denote the integer part of $y$. As $x\to\infty$, consider the sum

$$\sum_{p\le x}\phi\biggl(\biggl[\frac{x}{p}\biggr]\biggr).$$

Asymptotic formula.

$$\sum_{p\le x}\phi\biggl(\biggl[\frac{x}{p}\biggr]\biggr)\sim \frac{1}{\zeta(2)}x\log\log x.$$

The preceding theorem gives upper and lower bounds with constants differing from $1/\zeta(2)$ by fixed multiples of $1-1/\zeta(2)$ and $1/\zeta(2)$, respectively. The authors state that proving the asymptotic formula is currently out of reach because of difficulties in estimating the relevant exponential sums.