The asymptotic complexity conjecture for arithmetic expressions

Let $O=\{1,S,+,*\}$ and $O_{|}=\{1,S,+,*\}$ denote the two operation sets considered, with $c_O(n)$ and $c_{O_{|}}(n)$ their corresponding complexities. The logarithms are taken in the indicated bases.

Asymptotic complexity conjecture.

$$\frac{c_{O}(n)}{5\log_4(n)}\to 1\quad\text{as }n\to\infty$$

Alternatively,

$$\frac{c_{O_{|}}(n)}{3\log_3(n)}\to 1\quad\text{as }n\to\infty.$$

Results cited in the source give competing computational and experimental indications about the eventual coefficient, so the long-term behavior remains unresolved.