Linearity conjecture for small minimal blocking sets

Prove that every minimal blocking set B in PG(2,q) of size less than 3(q+1)/2 is a linear blocking set.

Background

The paper invokes the linearity conjecture while transferring results about blocking sets in projective planes to Cameron–Liebler sets on the Klein quadric. Minimal blocking sets of size below 3(q+1)/2 are known to satisfy strong intersection-number restrictions, but the cited conjecture asserts the stronger structural conclusion that all such sets are linear.

This conjecture is relevant because projected subgeometries and Baer subplanes provide examples of linear blocking sets, and their corresponding point-plane configurations yield Cameron–Liebler sets under the Klein correspondence.

References

The linearity conjecture states that every minimal blocking set $B$ in $\PG(2,q)$, of size $|B|< 3(q+1)/2$, is a linear blocking set, see .

— Cameron-Liebler sets of generators in the Klein quadric $Q^+(5,q)$  (2503.08260 - D'haeseleer et al., 11 Mar 2025) in Section 5, paragraph listing classical examples of non-trivial minimal blocking sets