Linearity of small minimal blocking sets

Prove that every minimal blocking set B in PG(2,q) with |B|<3(q+1)/2 is a linear blocking set.

Background

The paper uses results on blocking sets in PG(2,q) to transfer characterisation theorems about holes of maximal partial line spreads to Cameron–Liebler sets on the Klein quadric. Small minimal blocking sets are particularly relevant because their structure controls the possible configurations of holes and hence the corresponding Cameron–Liebler constructions.

The stated linearity conjecture concerns all minimal blocking sets below the size threshold 3(q+1)/2. Establishing it would imply that every such blocking set arises from a linear construction, substantially restricting the configurations that can occur in the associated geometric and Cameron–Liebler problems.

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, subsection “Link with holes of a maximal partial spread,” item 3 in the list of examples of non-trivial minimal blocking sets