Functions on a convex set which are both $ \omega $-semiconvex and $ \omega $-semiconcave II

In a recent article (2022) we proved with L. Zaj\'i\v{c}ek that if $G\subset\R^n$ is an unbounded open convex set that does not contain a translation of a convex cone with non-empty interior, then there exist $f:G\to\R$ and a concave modulus $\omega$ such that $\lim_{t\to\infty}\omega(t)=\infty$, $f$ is both semiconvex and semiconcave with modulus $\omega$ and $f\notin C^{1,\omega}(G)$. Here we improve the previous result as follows: If $G$ is as above and $\omega(t)=t^{\alpha}$ for some $\alpha\in(0,1)$, then there exists $f:G\to\R$ that is both semiconvex and semiconcave with modulus $\omega$ and $f\notin C^{1,\alpha}(G)$. This result has immediate consequences concerning a first-order quantitative converse Taylor theorem and the problem whether $f\in C^{1,\alpha}(G)$ whenever $f$ is smooth in a corresponding sense on all lines.