Let $(X,\rho)$ be a metric space. For any $f:X\to\mathbb{R}$, define the local Lipschitz constant of $f$ at $x$ by $$ \Lambda_f(x) := \sup_{x'\in X\setminus\{x\}} \frac{|f(x)-f(x')|}{\rho(x,x,')} . $$
I conjecture that for every continuous $f:X\to[0,1]$ and $\eta>0$ there is an $h:X\to[-1,1]$ such that $\overline f:=f+\eta h$ satisfies $$ \Lambda_{\overline{f}}(x) \le \Lambda_f(x) $$ for all $x\in X$ and $$ |\Lambda_{\overline f}(x)-\Lambda_{\overline f}(x')| \le \eta^{-1}\rho(x,x')\max\{\Lambda_f(x),\Lambda_f(x')\}^2 $$ whenever $x,x'\in X$ and at least one of $\Lambda_{\overline f}(x),\Lambda_{\overline f}(x')<\infty$.
Question: Are results of this kind known? Conversely, is there an inherent reason why the above (or some close variant) cannot possibly hold? A brief search seems to indicate that results in this general spirit are obtained here: https://arxiv.org/pdf/1812.11087.pdf ; am I at least correct about the "general spirit" part?
