Conservation of Ramsey's theorem for pairs and well-foundedness

In this article, we prove that Ramsey’s theorem for pairs and two colors is ${\mathrm{\Pi }}_1^1$-conservative over~${\mathsf{R}\mathsf{C}\mathsf{A}}_0+\mathsf{B}{\mathrm{\Sigma }}_2^0+\mathsf{W}\mathsf{F}({\epsilon }_0)$ and over~${\mathsf{R}\mathsf{C}\mathsf{A}}_0+\mathsf{B}{\mathrm{\Sigma }}_2^0+{\bigcup }_n\mathsf{W}\mathsf{F}({\omega }_n^{\omega })$. These results improve theorems from Chong, Slaman and Yang and Kołodziejczyk and Yokoyama and belong to a long line of research towards the characterization of the first-order part of Ramsey’s theorem for pairs.