Saturation conjecture for matrix preorderings on unions of unbounded intervals and a point

Let $K\subseteq \mathbb R$ be either a union of an unbounded interval and a point, or a union of two unbounded intervals and a point. Suppose $S$ is the natural description of $K$. Saturation conjecture. The $n$-th matrix preordering $T_S^n$ is saturated for every natural number $n>1$. This conjecture concerns the remaining cases for unions of one or two unbounded intervals and a point not covered by the preceding results; it is based on the investigation of examples, and no resolution is given here.