The corrected Welsh conjecture on the possible numbers of matroid bases

Let $n$ and $r$ be integers, and let $b$ be the number of bases of a matroid of rank $r$ on $n$ elements. The necessary bounds are

$$1 \leq b \leq \binom{n}{r}.$$

Corrected Welsh conjecture. For every triple $(n,r,b)$ such that $0 \leq r \leq n$ and $1 \leq b \leq \binom{n}{r}$, there is a matroid of rank $r$ on $n$ elements with exactly $b$ bases, except when

$$(n,r,b)=(6,3,11).$$

The database computation found no other missing triples for $n\leq 9$, motivating this corrected formulation; the conjecture remains open beyond the computed range.