Extremal basis count for circumference-bounded matroids

Determine whether, for all integers s at least 4 and r at least s-1, the maximum number of bases of an n-element rank-r matroid with no U_{s,s+1}-minor is at most binom{n/(r/(s-1))}{s-1}^{r/(s-1)}, with equality under the stated divisibility conditions for the direct sum of r/(s-1) copies of U_{s-1,n/(r/(s-1))}.

Background

The paper proves the corresponding extremal behavior for s=2 and s=3. These cases suggest that, among matroids with circumference at most s, the extremal construction should be a direct sum of uniform matroids of rank s-1.

The conjecture extends this pattern to every s at least 4 and specifies both the proposed upper bound and the equality construction. The authors note that their direct structural argument for the s=3 case does not seem to apply, although they expect the conjecture may not be difficult to prove.

References

We next consider $s \ge 4$. Matroids with circumference $s$ are not as well-structured when $s \ge 4$, but we expect that $\ex_{M}(n,r,U_{s,s+1})$ is achieved by a direct sum of uniform matroids of rank $s - 1$, as it is for $s = 2$ and $s = 3$.

— Turán densities for matroid basis hypergraphs  (2502.03673 - Pol et al., 5 Feb 2025) in Conjecture 5.1, Section 5 (U_{s,s+1})