 Conjugacy of positive real-line bumps

Choose fundamental intervals $[x,xf)$ and $[y,yg)$ for two positive bumps of an order automorphism. Any increasing bijection matching their endpoints extends by $b(x{f}^n)=(xb)g^n$ to their support intervals. Extend it monotonically over the two complementary rays. Then $fb=bg$, so $b^{-1}fb=g$. Bounded supports impose no further invariant for positive bumps in the full real-line order-automorphism group.