Maximal real count for the QL² constituent problem

Determine the maximal number of real solutions of the QL² constituent enumerative problem arising in the mixed degeneration of three conics, in which circles are tangent to one irreducible conic and two lines, and establish whether the associated degenerate configuration can be perturbed while preserving the resulting real solution count.

Background

The paper studies a mixed degeneration in which one conic remains irreducible while the other two conics degenerate to double lines. The resulting degree-184 enumerative problem decomposes into four constituent problems, denoted QP², QPL, QLP, and QL². The QL² problem concerns circles tangent to one conic and two lines, and contributes four times its real solution count to the total count for the original problem.

Prior work had established that the QL² constituent problem could have at least 12 real solutions, while the paper exhibits a configuration with 14 real solutions. However, the paper does not determine the maximal possible real count for QL², nor does it establish whether the degenerate configuration can be perturbed while preserving the relevant real count. Consequently, the paper identifies this as an unresolved problem despite producing 144 real tritangent circles overall.