Asymptotic counting conjecture for powers of multiplicatively independent complex numbers

Let $\alpha,\beta\in\mathbb{C}$ satisfy $|\alpha|>1$ and $|\beta|>1$. Two nonzero complex numbers are multiplicatively independent if no nontrivial integer relation $\alpha^r\beta^s=1$ holds with $(r,s)\in\mathbb{Z}^2\setminus\{(0,0)\}$. Define

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

The multiplicative-independence counting conjecture. The numbers $\alpha$ and $\beta$ are multiplicatively independent if and only if

$$N_{\alpha,\beta}(x)\sim\frac{(\log x)^2}{\log|\alpha|\cdot\log|\beta|},\qquad x\to\infty.$$

This is proposed as an extension of the paper's asymptotic results from algebraic to potentially transcendental $\alpha$ or $\beta$; its resolution is not supplied here.