The iterated-logarithm infinity conjecture

Let $f(x)$ be a function such that

$$f(x)|_{x=d4a}=d4a,$$

and let $k=d4a$ denote an infinite iteration index. Iterated-logarithm infinity conjecture. When $x$ reaches infinity before $k$, one has

$$\operatorname{ln}_{k} f(x)=d4a.$$

The claim concerns the behavior of infinitely nested logarithms in the paper's infinitary calculus. The source gives no proof or resolution.