Optimality of the isosceles trapezoid inscription theorem

Let $\gamma_1,\gamma_2\colon\mathbb{R}\to\mathbb{C}$ be continuous disjoint periodic embeddings of the real line into the plane. A quadrilateral $Q$ admits an inscription in any pair of disjoint periodic curves if every such pair contains four points forming a quadrilateral similar to $Q$. Optimality conjecture. If $Q$ is a quadrilateral that admits an inscription in any pair of disjoint periodic curves in $\mathbb{C}$, then $Q$ is an isosceles trapezoid. The theorem in the paper establishes universal inscription for every isosceles trapezoid, while the conjecture asserts that no broader class of quadrilaterals has this property.