On a Problem of Janusz Matkowski and Jacek Weso{l}owski, II

We continue our study started in "On a problem of Janusz Matkowski and Jacek Weso{\l}owski" (see arXiv:1703.08459) of the functional equation \begin{equation*} \varphi(x)=\sum_{n=0}^{N}\varphi(f_n(x))-\sum_{n=0}^{N}\varphi(f_n(0)) \end{equation*} and its increasing and continuous solutions $\varphi\colon[0,1]\to[0,1]$ such that $\varphi(0)=0$ and $\varphi(1)=1$. In this paper we assume that $f_0,\ldots,f_N\colon[0,1]\to[0,1]$ are strictly increasing contractions such that \begin{equation*} 0\leq f_0(0)<f_0(1)\leq f_1(0)<\cdots <f_{N-1}(1)\leq f_N(0)<f_N(1)\leq 1 \end{equation*} and at least one of the weak inequalities is strong.