The maximal ball-number conjecture for finite ordinal spaces

An ordinal space is a space $(X,\delta)$ with its associated set of balls $\mathbf B_X$. Let $|X|=n\geqslant 1$, and write $b_n$ for the maximal possible value of $|\mathbf B_X|$ among ordinal spaces of cardinality $n$. The proposed values begin

$$b_1=1,\quad b_2=3,\quad b_3=6,\quad b_4=12,\quad b_5=19,\quad b_6=29,\quad b_7=40,\ldots$$

Maximal ball-number conjecture. The maximal number of balls in $X$ is $b_n$, where $(b_n)$ is sequence A263511. The paper supports the sequence by direct constructions, including one with $n=6$, but explicitly says that such a construction is not a proof of maximality; the conjecture remains unresolved in the supplied text.