Exact Turan basis number below the projective-geometry threshold

Determine whether, for a prime power t>2 and integers 1<=c<=r-1, the maximum number of bases in an (t^r-t^{r-c})/(t-1)-element, rank-r matroid with no U_{2,t+2}-minor equals the number of bases of the matroid obtained by restricting PG(r-1,t) to the complement of a rank-(r-c) flat.

Background

The paper establishes the extremal basis count for U_{2,t+2}-minor-free matroids when the ground-set size is sufficiently large and, for prime-power t, when the size is a multiple of the number of points in PG(r-1,t).

It then proposes an exact formula for smaller ground sets obtained by deleting a flat from the projective geometry. The question asks whether this Bose–Burton-type construction remains extremal in that regime.