= Solution
Write the two supports as $I_f=(l_f,r_f)$ and $I_g=(l_g,r_g)$. Every point in a support moves strictly right: a positive nonidentity <order automorphism> has no fixed point inside its one support interval. Choose $u\in I_f$, $v\in I_g$ and an increasing bijection $\theta:[u,uf]\to[v,vg]$ matching endpoints. The half-open intervals $[uf^n,uf^{n+1})$, for $n\in\mathbb Z$, partition $I_f$, because the iterates tend to its endpoints. Define
$$
(tf^n)b=(t\theta)g^n,\qquad u\le t<uf.
$$
The endpoint choices make adjacent pieces fit, giving an increasing bijection $b:I_f\to I_g$ with $fb=bg$. Extend over the two complementary closed rays by increasing maps taking $l_f,r_f$ to $l_g,r_g$. Outside the supports both automorphisms are identity, so $fb=bg$ still holds. Therefore $\boxed{b^{-1}fb=g}$. This gives the <conjugacy of positive real-line bumps>.
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