Euler's totient formulation of the binary Goldbach conjecture

Let $\phi$ denote Euler's totient function. For every natural number $n$ distinct from unity, seek integral solutions $x$ to the displayed system

$$\begin{cases} \phi(n-x)+1=n-x \\ \phi(n+x)+1=n+x \end{cases}$$

Euler's totient formulation of the binary Goldbach conjecture. The system has an integral solution for every natural number $n$ distinct from unity. This is presented as an alternative formulation of the binary Goldbach conjecture, which asserts that every even integer $2n$, $n>1$, is a sum of two primes. The supplied text does not establish the conjecture itself or provide evidence that this formulation has been resolved.