Conjugating positive bumps while preserving a fundamental interval

If two positive bumps of an order automorphism move a point $c$ beyond an interval $[c,d]$, their conjugacy can be chosen to fix that interval: begin with the identity there, complete the increasing map between fundamental intervals, and extend equivariantly along bump iterates. If both bump supports lie in one fundamental interval of another positive motion $h$, fix that cell's endpoints and repeat the conjugator on all its $h$-translates. The result commutes with $h$, and with any automorphism supported in the interval fixed pointwise.