Chen–Chvátal conjecture for finite metric spaces

Let $(V,d)$ be a finite metric space with $n$ points. For distinct points $x,y,z\in V$, say that $z$ lies between $x$ and $y$ when

$$d(x,y)=d(x,z)+d(z,y).$$

For distinct $x,y\in V$, define the line generated by $x$ and $y$ by

$$\overline{xy}=\{x,y\}\cup\{z\in V: d(z,y)=d(z,x)+d(x,y)\ \lor\ d(x,y)=d(x,z)+d(z,y)\ \lor\ d(x,z)=d(x,y)+d(y,z)\}.$$

A metric space has a universal line if its whole point set is a line. Chen–Chvátal conjecture. Every finite metric space of $n$ points without a universal line determines at least $n$ distinct lines. The conjecture extends the de Bruijn–Erdős theorem from the Euclidean plane to arbitrary finite metric spaces and remains open in general, although it is known for graphs of diameter three.