The field-equivalence problem for transcendental-curve intersections

Consider the fields obtained by allowing general intersections with the transcendental curves mentioned in the source, namely trigonometric functions, the Quadratrix of Hippias, and the Archimedean spiral. Transcendental-curve field conjecture. These fields are identical, and possibly form a larger field denoted $\mathcal{T}_2$. The source gives no resolution and presents this as an unresolved question about general intersections with transcendental curves.