The infinite-interval L-function comparison conjecture

Let $f$ and $g$ be L-functions, let $a,b\in\Phi^{-1}$, and suppose $b-a=\pmd4a$. For $z\in\{\prec,\succ,\propto\}$, compare the functions over the interval $[a,b]$.

Infinite-interval comparison conjecture. One has

$$f\;z\;g|_{x=[a,b]}\equiv f\;z\;g|_{x=d4a}.$$

This extends the paper's comparison theory from evaluation at infinity to comparison over an infinite interval. No proof or resolution is supplied.