Conjectured reflection classifications for labelled Kruskal's theorem

Let $\operatorname{KT}_{\ell}(\omega)$ denote labelled Kruskal's theorem with labels drawn from $\omega$, and let $\operatorname{KT}_{\ell}(n)$ denote its restriction to labels below $n$. Let $\Pi^1_2\text{-}\omega\operatorname{RFN}(\Pi^1_2\text{-}\operatorname{BI}_0\upharpoonright\Pi^1_3)$ and $\Pi^1_2\text{-}\operatorname{RFN}(\Pi^1_2\text{-}\operatorname{BI}_0)$ be the corresponding uniform reflection principles over the indicated base theories. Pakhomov–Freund conjectures. The following classifications hold:

$$\textsf{RCA}_0\vdash\operatorname{KT}_{\ell}(\omega)\leftrightarrow \Pi^1_2\text{-}\omega\operatorname{RFN}(\Pi^1_2\text{-}\operatorname{BI}_0\upharpoonright\Pi^1_3),$$

and

$$\textsf{RCA}_0\vdash\forall n\,\operatorname{KT}_{\ell}(n)\leftrightarrow \Pi^1_2\text{-}\operatorname{RFN}(\Pi^1_2\text{-}\operatorname{BI}_0).$$

These conjectures seek proof-theoretic classifications of labelled Kruskal's theorem and its finite-label versions in terms of reflection principles, extending the known classification of unlabelled Kruskal's theorem. The first proposed classification is attributed to F. Pakhomov and the second to A. Freund; their status is not established in the supplied source.