A conjecture by Chen and Chvátal asks for the minimum number of induced "lines" in a metric space, in the same spirit as the De Bruijn–Erdős theorem.

Though the statement of this problem on Douglas West's page on the conjecture asked about lines, I was wondering if any work had been done on the problem with maximal lines (which are not a proper subset of another line) but lack the resources to check. It would be easier to disprove than the original conjecture, but I was wondering if there was an easy counterexample before hitting it over the head with a computer?