Maximality conjecture for real circles tangent to three conics

Let $Q_1,Q_2,Q_3\subseteq\mathbb{R}^2$ be three general real conics. A real circle is a circle defined over $\mathbb{R}$ that is tangent to each of $Q_1,Q_2,Q_3$. Maximality conjecture. The maximal number of real circles tangent to three conics is $136$. The paper constructs an instance with $136$ such circles, but does not prove that this number is maximal; computational evidence and a degenerate construction motivate the conjecture.