let $\mathcal{P}$ be a finite set of points in the euclidean plane in general position, and let $$\lbrace p_A,p_B,p_C,p_D\rbrace: \ \|p_C-p_A\|+\|p_D-p_B\|\ \gt \|p_i-p_h\|+\|p_k-p_j\|\quad\forall\ \mathcal{P}\supset\lbrace h,i,j,k\rbrace\ne\lbrace A,B,C,D\rbrace\subset\mathcal{P}$$ be the quadrilateral with the maximal induced maximum weight perfect matching.

Question:Is it possible, that $\left( p_A,p_C\right)$ or $\left(p_B,p_D\right)$ is contained in the shortest tour through the points of $\mathcal{P}$?

I am convinced, that such a shortest tour can't exist, but don't know, how to go about proving that; therefore I am hoping for a proof proving or disproving the existence of such optimal tours.