Nonexpansive, $\mathcal{T}$-continuous antirepresentations have common fixed points

Let $C$ be a closed convex subset of a Banach (dual Banach) space $\mathfrak{X}$. By $\mathcal{S}$ we denote an antirepresentation $\{ T_{s} \,:\, s \in S \}$ of a semitopological semigroup $S$ as nonexpansive mappings on $C$. Suppose that the mapping $S \times C \ni (s,x) \to T_{s}x \in C$ is jointly continuous when $C$ has the weak (weak*) topology and the Banach space $RUC(S)$ of bounded right uniformly continuous functions on $S$ has a right invariant mean. If $C$ is weakly compact (for some $x \in C$ the set ${\overline{\{ T_{s}x \,:\, s \in S \}}} ^{\text{weak*}}$ is weakly* compact) and norm separable, then $\{ T_{s} \,:\, s \in S \}$ has a common fixed point in $C$.