For every choice of reals $$x_1 \leq \cdots \leq x_n, \quad y_1 \leq \cdots \leq y_n$$ and every permutation $$x_{\pi(1)},...,x_{\pi(n)},$$ it is known that $$x_ny_1 + \cdots + x_1y_n \leq x_{\pi(1)}y_1 + \cdots + x_{\pi(n)}y_n \leq x_1y_1+ \cdots +x_n y_n.$$ This is also referred to as the rearrangement inequality.
Suppose we consider a slightly modified problem as follows. Consider we have two weighted, undirected, complete graphs $A=(V,X), B=(V,Y)$ with the same node set $V$. Edge weights satisfy triangle inequality. The number of edges is $|E| = |V|(|V|-1)/2$. Note that $X=(x_{ij})$ and $Y=(y_{ij})$ are adjacency matrices. Then we have two sorted sequences of weights, one for each graph: $$x^1\leq\cdots\leq x^{|E|}, \quad y^1 \leq \cdots \leq y^{|E|} .$$ Suppose we split each sequence into two sub-sequences such that one contains only edge weights in its maximum spanning tree, the other consists of the rest of the weights: $$x^{(1)}\leq\cdots\leq x^{(|V|-1)}, \quad x^{(|V|)}\leq\cdots\leq x^{(|E|)}$$ and $$y^{(1)}\leq\cdots\leq y^{(|V|-1)}, \quad y^{(|V|)}\leq\cdots\leq y^{(|E|)}.$$
My question: Is the statement below true for all $A$ and $B$? $$ \sum_{i=1}^{|V|} \sum_{j>i} x_{ij}y_{ij} \leq (x^{(1)} y^{(1)} + \cdots + x^{(|V|-1)} y^{(|V|-1)}) + (x^{(|V|)} y^{(|V|)} + \cdots + x^{(|E|)} y^{(|E|)})$$
If not, what is the upper bound of the variable $C$? $$ C \sum_{i=1}^{|V|} \sum_{j>i} x_{ij}y_{ij} \leq (x^{(1)} y^{(1)} + \cdots + x^{(|V|-1)} y^{(|V|-1)}) + (x^{(|V|)} y^{(|V|)} + \cdots + x^{(|E|)} y^{(|E|)}) .$$
Thank you for the help!