Asymptotic counting conjecture for differences of powers of $\pi$ and $e$

For $x>0$, define

$$N_{\pi,e}(x)=\#\{(n,m):|\pi^n-e^m|\leqslant x,\ (n,m)\in\mathbb{N}\times\mathbb{N}\}.$$

The $\pi$–$e$ counting conjecture. As $x\to\infty$,

$$N_{\pi,e}(x)\sim\frac{(\log x)^2}{\log\pi}.$$

The authors state that this conjecture can imply the irrationality of $\log(\pi)$; no resolution is given in the source.