Reflection-principle classifications for labelled Kruskal's theorem

Let $\text{KT}_{\ell}(\omega)$ denote the labelled Kruskal theorem for trees labelled by natural numbers, and let $\text{KT}_{\ell}(n)$ denote its restriction to labels below $n$. Write $\Pi^1_2\text{-}\omega\operatorname{RFN}(\Pi^1_2\text{-}\operatorname{BI}_0\upharpoonright\Pi^1_3)$ for the relevant $\omega$-reflection principle, and $\Pi^1_2\text{-}\operatorname{RFN}(\Pi^1_2\text{-}\operatorname{BI}_0)$ for the corresponding uniform reflection principle. Here $\textsf{RCA}_0$ is the base theory of recursive comprehension.

Pakhomov–Freund conjectures. The following two equivalences are conjectured:

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

and

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

These proposed classifications aim to determine the proof-theoretic strength of labelled Kruskal's theorem and of its finite-label versions, extending the known classification of unlabelled Kruskal's theorem by Rathjen and Weiermann. The source attributes the first equivalence to F. Pakhomov and the second to A. Freund; no resolution is given here.