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Construct the conjugacy of positive real-line bumps using $u=v=\alpha$. Since both $\alpha f$ and $\alpha g$ are greater than $\beta$, choose the fundamental-interval map to be identity on $[\alpha ,\beta ]$ and then increase from $\beta$ to the respective right endpoints. Equivariant extension along $f$ and $g$ gives a conjugator on their supports fixing $[\alpha ,\beta ]$. Extend it over the remaining portions of $J=[\gamma ,\gamma h]$, fixing the endpoints of $J$; both support intervals lie strictly inside $J$. Call the resulting interval bijection $b_0$. It satisfies $f{b}_0=b_{0}g$ on $J$.

The point $\gamma$ lies in a positive orbital of $h$, because $\gamma h>\gamma$. Its half-open cells $J{h}^n$, $n\in \mathbb{Z}$, partition that orbital. Extend byand set $b$ to identity outside the orbital. Since $b_0$ fixes the cell endpoints, this is an order automorphism and satisfies $bh=hb$. Each cell is preserved; $f$ and $g$ are identity on all cells other than the original one, so $fb=bg$ holds everywhere. Finally $b$ fixes $[\alpha ,\beta ]$ pointwise, and $a$ is supported inside that interval. Hence $ba=ab$. ThusThis is conjugating positive bumps while preserving a fundamental interval; merely choosing an arbitrary conjugator would not ensure either commutation relation.