Let $A$ be an unbounded linear operator of domain $D(A)$ defined on a Banach space $X$. Suppose that $A$ generates a $C_0$-semigroup $T(t)$ which is uniformly bounded. I would like to know if there are additional assumptions under which the following conjecture is true:

For every compact subset $K$ of $X$, there exists a constant $C_K$ such that $$\mid{((T(t)-T(t'))x\mid \leq C_K\mid t-t'\mid},\quad \forall t,t'\in \mathbb{R}_+,\quad \forall x\in K.$$