The minimal ball-number conjecture for finite ordinal spaces

Let $(X,\delta)$ be an ordinal space with $|X|=n$, and let $\mathbf B_X$ denote its set of balls. Assume that

$$\delta(x,y)\neq\delta(z,w)$$

for every different pairs of points $\{x,y\}\neq\{z,w\}$ with $x\neq y$ and $z\neq w$. Minimal ball-number conjecture. The following inequality holds:

$$\frac{n(n+1)}{2}\leqslant |\mathbf B_X|.$$

Equality holds if and only if the Hasse diagram $\mathcal H(X)$ has the structure depicted in Figure 2. Without the distinct-distance restriction, the paper notes that the minimum is trivially $|X|+1$ when $|X|>1$; the restricted lower bound and its equality characterization are presented as a conjecture, with no resolution supplied.