The L-function comparison conjecture

An L-function in $*G$ is a finite combination of the operations $\{+, -, \div, \times, \operatorname{ln}, \operatorname{e}\}$. For functions compared at infinity, write $f\;z\;g|_{n=d4a}$.

L-function comparison conjecture. Every L-function at infinity is ultimately continuous and monotonic. If $f$ and $g$ are L-functions, then in

$$f\;z\;g|_{n=d4a},$$

$z$ is unique and

$$z\in\{\prec,\succ,\propto\}.$$

The statement is presented as a conjectural reformulation of a result attributed to Hardy, because L-functions have been redefined in $*G$ rather than over the real numbers. The source provides no resolution of the reformulated claim.