The infinite-integer ratio transfer conjecture

Let $\mathbb{N}_{\infty}$ denote the infinite integers. Infinite-integer ratio transfer conjecture. There exist ratios of infinite integers of the form

$$\frac{\mathbb{N}_{\infty}}{\mathbb{N}_{\infty}}$$

that can be transferred to real numbers.

The claim is motivated by the paper's discussion of rational finite approximations whose values at infinity may be promoted to other number types, including transcendental real numbers. The source provides no proof or resolution.