Does $|T(f) - T(g)| \leq |T(f-g)|$ hold for a sublinear operator $T$?

Question

Let $X,Y$ be function spaces with functions taking values in $\mathbf{C}$. An operator $T:X\to Y$ is called sublinear if for all $f,g \in X$ and all $\lambda \in \mathbf{C}$, we have $$ |T(\lambda f)| = |\lambda| |T(f)| $$ and $$ |T(f+g)| \leq |T(f)| + |T(g)|. $$ My question is if any such operator also satisfies $$|T(f) - T(g)| \leq |T(f-g)|$$ for any functions $f,g \in X$. If not, what is an example in which this fails? I'm thinking it is not true, but I have not been successful in constructing such an example. The reason we're interested in such an inequality is that if it is true in general, then we can prove a certain operator defined on characteristic functions can be extended to all functions.

Answer 1

No. Consider $$ T(f)=\Vert f\Vert e^{i \Vert f\Vert} $$ and constant function $f=\pi$ and $g=2\pi$.