Classification conjecture for monotone proof-theoretic operators

A monotone proof-theoretic operator is a monotone function sending sentences to sentences; a function is recursive when it is computable, and it is non-constant when it is not constant. Let $T$ be the fixed base theory, let $\beta\preceq\alpha$ denote the ordering between ordinal notations, and let $[\theta]$ denote the equivalence class of a sentence under the paper's proof-theoretic equivalence. For a nice elementary presentation $\prec$ of a recursive well-ordering, define the iterated consistency statements by

$$T \vdash \mathsf{Con}^\alpha_T(\varphi) \leftrightarrow \forall \beta \prec \alpha\,\mathsf{Con}_T\bigl(\varphi\wedge\mathsf{Con}^\beta_T(\varphi)\big).$$

Classification conjecture for monotone proof-theoretic operators. Suppose $\mathfrak{g}$ is monotone, non-constant, and recursive, with $\mathfrak{g}(\varphi)\in\Pi_1$ for every sentence $\varphi$. Let $\prec$ be a nice elementary presentation of a well-ordering and let $\alpha$ be an ordinal notation. If, for every $\varphi$,

$$T+\varphi+\mathsf{Con}^\alpha_T(\varphi)\vdash\mathfrak{g}(\varphi),$$

then there is some $\beta\preceq\alpha$ such that, for all $\varphi$ in a true cone,

$$[\varphi\wedge\mathfrak{g}(\varphi)]=[\varphi\wedge\mathsf{Con}^\beta_T(\varphi)].$$

The conjecture seeks a classification of monotone proof-theoretic operators analogous to the Slaman–Steel classification of increasing Borel order-preserving functions. It refines the preceding dichotomy for recursive monotone operators bounded by the consistency operator; the source provides no resolution, so the conjecture remains open.