Universal optimality of the logarithmic coordinate for heavy tails

Let $\mathcal{P}$ be a distribution family on $(0,\infty)$ whose members have power-law tails satisfying $P(X>x)\sim x^{-\alpha}$ for some $\alpha>0$. Let the logarithmic coordinate be $\psi(x)=\log x$. Universal optimality conjecture. The coordinate $\psi$ is asymptotically optimal as the tail parameter $\alpha\to0$. The conjecture proposes a universal coordinate choice for increasingly heavy-tailed distribution families; the supplied source gives no evidence establishing or refuting it.