The monotone-operator classification conjecture

Let $T$ be a sound recursively enumerable extension of $\mathsf{EA}$. Let $\mathfrak{g}$ be a non-constant, recursive, monotone function such that $\mathfrak{g}(\varphi)$ is $\Pi_1$ for every $\varphi$. Let $\prec$ be a nice presentation of a well-ordering, and suppose that for every $\varphi$,

$$T+\varphi+\mathsf{Con}_T^\alpha(\varphi)\vdash\mathfrak{g}(\varphi).$$

Here $\mathsf{Con}_T^\alpha$ denotes the indicated iterate of the consistency operator, and $[\cdot]_T$ denotes the equivalence class used in the source.

Monotone-operator classification conjecture. For some $\beta\preceq\alpha$ and some true sentence $\varphi$, for every $\psi$ such that $T+\psi\vdash\varphi$,

$$[\psi+\mathfrak{g}(\psi)]_T=[\psi+\mathsf{Con}_T^\beta(\psi)]_T.$$

The conjecture would classify suitable recursive monotone operators by iterates of the consistency operator. The source presents it as a hoped-for classification following a positive answer to an earlier question; no resolution is given.