We are given $n$ points on a circle of radius $1$ and an even number $k$. The distance $d(x,y)$ between two points $x$ and $y$ on the circle is given by the length of the (shorter) arc connecting them in the circle. We consider two challenges for such an instance:
Finding a maximum matching with $k/2$ pairs, i.e. finding disjoint pairs of nodes $(s_1,t_1),\dots,(s_{k/2},t_{k/2})$ for which: $$ \sum_{i=1}^{k/2} d(s_i,t_i)$$ is maximized.
Finding a set of $k$ points with maximum sum of pairwise distances, i.e. a set $S$ of points with $|S| = k$ such that: $$ \sum_{x,y \in S} d(x,y) $$ is maximized.
Now assume we are given a maximum matching with $k/2$ pairs $(s_1,t_1),\dots,(s_{k/2},t_{k/2})$, i.e. an optimal/maximal solution for challenge 1. Let $S := \{s_1,t_1,\dots,s_{k/2},t_{k/2}\}$ be the set of the points used in the matching. Prove or disprove that the set $S$ has the optimal/maximal sum of pairwise distances among sets of size $k$, i.e. is optimal for challenge 2.
The problem is posed for general $k$, but a solution even only for $k=4$ would be interesting. Note that for the case $k=2$ the statement is quite obviously true.
I think I found a counterexample for $k=4$. Assume wlog that the center of the circle is at $(0,0)$ and that it has radius $1 / \pi$. I identify a point $p := (x,y)$ on the circle by the length of the arc going from $(1/\pi,0)$ to its position $(x,y)$. Hence all points on the circle are identified with numbers from $[0,2)$
Consider the $5$ points: $p_1 = 1.98$, $p_2 = 0.89$, $p_3 = 1.97$, $p_4 = 0.72$, $p_5 = 1.42$.
The maximum matching is given by $(p_1,p_2)$, $(p_3,p_4)$ with size $1.66$. The set $\{p_1,p_2,p_3,p_4\}$ has sum of pairwise distances $3.5$.
This is not optimal: the set $\{p_2,p_3,p_4,p_5\}$ has sum of pairwise distances $3.66$. The maximum matching induced by it, given by $(p_2,p_3)$, $(p_4,p_5)$ has size $1.62 < 1.66$.