Interval maps quasi-symmetrically conjugate to a piecewise affine map

Consider a multimodal interval map f of C3 with non-flat critical points. We establish several characterizations of the map f that is quasi-symmetrically conjugate to a piecewise affine map in the case f is topologically exact and all of its periodic points are hyperbolic repelling. In particular, we give a negative answer to a question posed by Henk Bruin in [1].