Nonuniform computability conjecture for special fan functionals and Pincherle realisers

Let $M_{0}^{3}$ be a type-three functional satisfying the Pincherle realiser condition $\textup{\textsf{PR}}(M_{0})$, and let $\Theta^{3}$ be a type-three functional satisfying $\textup{\textsf{SCF}}(\Theta)$. Computability is in the S1–S9 sense.

Nonuniform computability conjecture. There is an $M_{0}^{3}$ satisfying $\textup{\textsf{PR}}(M_{0})$ such that no $\Theta^{3}$ satisfying $\textup{\textsf{SCF}}(\Theta)$ is computable in $M_{0}^{3}$.

This conjecture captures a proposed fundamental difference between special fan functionals and Pincherle realisers: although the corresponding principles are equivalent in suitable higher-order systems, the functional witnessing the special fan theorem may not be uniformly computable from a Pincherle realiser, even with additional comprehension. The source presents the claim as conjectural and notes that it may be incorrect.