Conjecture on the existence of linear matroids with prescribed numbers of bases

Let $(n,r,b)$ be integers with $0<r\leq n$ and $1\leq b\leq\binom{n}{r}$. A linear $(n,r,b)$-matroid is a matroid with $n$ elements, rank $r$, and exactly $b$ bases that is represented by a matrix over a field. Linear matroid existence conjecture. A linear $(n,r,b)$-matroid exists for all $0<r\leq n$ and $1\leq b\leq\binom{n}{r}$ except $(n,r,b)=(6,3,11)$. The paper presents this as a strengthening of the arbitrary-matroid conjecture. It establishes substantial ranges, but the complete assertion is presented as unresolved; the source explicitly discusses further asymptotic work toward a complete solution.