= Conjugating positive bumps while preserving a fundamental interval
Let $f,g$ be <positive bumps of an order automorphism> whose <supports of a permutation> have matching endpoint types, as required by <conjugacy of positive real-line bumps>. Suppose $c<d<\min\{cf,cg\}$, using right actions. Their <conjugacy> can be chosen to fix $[c,d]$ pointwise.
Start with an increasing <bijection> $b_0:[c,cf]\to[c,cg]$ that is the identity on $[c,d]$. Such an extension exists because both right endpoints lie strictly beyond $d$. Extend $b_0$ along the bump iterates using $b(tf^n)=b_0(t)g^n$, and extend over the complementary rays. Endpoint agreement makes the iterated pieces fit, and gives $fb=bg$ while retaining the prescribed fixed interval.
There is also a useful commuting version. If both bump supports lie strictly inside one fundamental interval of another <order automorphism> $h$ moving points to the right, choose the conjugator within that cell and fix its endpoints. Repeat it on all $h$-translates of the cell, and use the identity outside the resulting orbital. The pieces define a conjugator commuting with $h$. It also commutes with every <automorphism> whose support lies in an interval that the conjugator fixes pointwise.
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