Iterative roots of type $ \mathcal {T}_2 $

<p style='text-indent:20px;'>This paper aims to an open problem on iterative roots of PM functions, a class non-monotonic functions. The asks: Does function nonmonotonicity height <inline-formula><tex-math id="M2">\begin{document}$ \ge 2 $\end{document}</tex-math></inline-formula> have continuous root order id="M3">\begin{document}$ n being less than or equal the number forts? It was proved that id="M4">\begin{document}$ forts (if exist) can be classified into two types: mostly increasing ones (type id="M5">\begin{document}$ \mathcal {T}_1 $\end{document}</tex-math></inline-formula>) and decreasing id="M6">\begin{document}$ {T}_2 all type id="M7">\begin{document}$ are found, but remaining id="M8">\begin{document}$ is left for more complicated construction. In this full description id="M9">\begin{document}$ given.</p>