Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2

It is known that any piecewise monotonic function of nonmonotonicity height not less than 2 has no continuous iterative roots of order greater than the number of forts of the function. So the following problem arises naturally: does such a function have an iterative root of order nn not greater than the number of forts? We consider the case that the number of forts is equal to nn, in which there appear possibly only two types T1T1 and T2T2 of iterative roots, i.e., the roots strictly increasing on the interval stretched on all forts of the given function and the roots strictly decreasing on such an interval, respectively. We characterize all type T1T1 roots of order nn and give a necessary condition for a root of order nn to be of type T2T2. A full description of type T2T2 is still an open question.