Let $ABCD$ be a regular tetrahedron with center $O.$ Consider two points $M,N,$ such that $\overrightarrow{NO}=-3\overrightarrow{MO}.$ Prove or disprove that $$NA+NB+NC+ND\geq MA+MB+MC+MD$$

I tried to use CS in the Euclidean space $E_3$, but it does not help, because the minoration is too wide.
Note: I also posted this on the Mathematics Stack Exchange, but not much progress has been made on this question. This is why I thought that posting here too would be all right (this problem is open in the sense that its proposer doesn't have a proof, so I guess it is fit the for this forum).
EDIT: The bounty expired, so this may be reopened.