The algebraic-power conjecture for transcendental-curve intersections

Let $\mathcal{T}_1$ be the construction field from the preceding discussion, and consider allowing general intersections with any of the transcendental curves under consideration. Transcendental-curve algebraic-power conjecture. Such general intersections may allow the construction of algebraic numbers that are impossible to construct in $\mathcal{T}_1$. The source states this as an unresolved question and offers no evidence deciding whether the asserted possibility holds.