Intertwining of maxima of sum of translates functions with nonsingular kernels

In previous papers we investigated so-called sum of translates functions $F(\mathbf{x},t):=J(t)+{\sum }_{j=1}^n{\nu }_{j}K(t-x_j)$, where $J:[0,1]\to \underset{‾}{\mathbb{R}}:=\mathbb{R}\cup \{-\mathrm{\infty }\}$ is a “sufficiently nondegenerate” and upper-bounded “field function”, and $K:[-1,1]\to \underset{‾}{\mathbb{R}}$ is a fixed “kernel function”, concave both on $(-1,0)$ and $(0,1)$, and also satisfying the singularity condition $K(0)={\lim }_{t\to 0}K(t)=-\mathrm{\infty }$. For node systems $\mathbf{x}:=(x_1,\dots ,x_n)$ with $x_0:=0\le x_1\le \cdots \le x_n\le 1=:x_{n+1}$, we analyzed the behavior of the local maxima vector $\mathbf{m}:=(m_0,m_1,\dots ,m_n)$, where $m_j:=m_j(\mathbf{x}):={\sup }_{x_j\le t\le x_{j+1}}F(\mathbf{x},t)$. Among other results we proved a strong intertwining property: if the kernels are also decreasing on $(-1,0)$ and increasing on $(0,1)$, and the field function is upper semicontinuous, then for any two different node systems there are $i,j$ such that $m_i(\mathbf{x})<m_i(\mathbf{y})$ and $m_j(\mathbf{x})>m_j(\mathbf{y})$. Here we partially succeed to extend this even to nonsingular kernels.