The optimal sub-Gaussian normalisation for randomised monotone functions

Let $\mathcal{M}$ denote the class of randomised monotone functions on $\mathbb{R}$ with values in $[0,1]$, and let $U_{\mathcal{M}}\text{\hspace{0.17em}⁣}:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ be the minimal function for which $$\mathbb{P}\{\sqrt{{\eta }_f}\underset{t\in \mathbb{R}}{\sup }\mid f_Z(t)-\text{Exf}{f}_Z(t)\mid \ge \epsilon \sqrt{U_{\mathcal{M}}({\eta }_f)}\}\le 2{\text{e}}^{-2{\epsilon }^2}$$ holds for every member $f_Z$ of $\mathcal{M}$ with finite effective sample size ${\eta }_f$ and every positive $\epsilon$. We prove that for every $x>1$, $$\mid \sqrt{U_{\mathcal{M}}(x)}-\sqrt{{\log }_{4}x}\mid \le 2\min \text{\hspace{0.17em}⁣}\{1,\frac{2\ln (\text{e}+\ln x)}{\sqrt{\ln x}}\}\mathrm{.}$$ The optimal adjustment $\sqrt{U_{\mathcal{M}}(x)}$ matches $\frac{1}{\sqrt{2\ln 2}}\sqrt{\ln x}$ for all $x>1$, with residuals bounded as above.