Non-smoothability for a class of groups of piecewise linear homeomorphisms of the interval

For a certain class of groups of piecewise linear homeomorphisms of the interval, we prove that they admit no sufficiently regular faithful action on the line. Building on previous work of Brum, Matte Bon, Rivas, and the author, the new ingredient is an observation from a recent work of Hyde and Tatch Moore, which allows to reduce the problem to the case of the circle, and then apply Herman–Yoccoz theory.