Unrestricted products of contractions in Banach spaces

Let $X$ be a reflexive Banach space such that for any $x \ne 0$ the set $$\{x^* \in X^*: \text {$\|x^*\|=1$ and $x^*(x)=\|x\|$}\}$$ is compact. We prove that any unrestricted product of of a finite number of $(W)$ contractions on $X$ converges weakly.