Optimality of the six-approximation for three facilities

Determine whether the strategyproof 6-approximation obtained by the Product-Gap mechanism for three-facility location on the real line is optimal.

Background

For k=3, the paper proves that Product-Gap is strategyproof in expectation and achieves a 6-approximation. The paper does not establish whether no randomized strategyproof mechanism can achieve a smaller approximation ratio, leaving the optimal approximation factor for three facilities unresolved.