Equicontractive weak separation property on the line does not imply convex finite type condition

Let $\{S_1,S_2,\dots ,S_n\}$ be an iterated function system on $\mathbb{R}$ with attractor $K$. It is known that if the iterated function system satisfies the weak separation property and $K=[0,1]$ then the iterated function system also satisfies the convex finite type condition. We show that the condition $K=[0,1]$ is necessary. That is, we give two examples of iterated function systems on $\mathbb{R}$ satisfying weak separation condition, and $0<{\dim }_H(K)<1$ such that the IFS does not satisfy the convex finite type condition.