  In this paper, we study the twist-conjecture for Coxeter systems and rigidity
of Coxeter systems up to finite twists. For Coxeter systems $(W,R)$ and
$(W,S)$, under the untangle-condition for conjugate subsets, we investigate
separations and type(I) and type(II) subsets of $R$ and $S$ and give an
equivalent condition of $R$ and $S$ that are conjugate up to finite twists. We
provide one direction of approach to solving the twist-conjecture and the
isomorphism problem for Coxeter groups of finite ranks.
