The homology of ${\mathrm{S}\mathrm{L}}_2$ of discrete valuation rings

Let $A$ be a discrete valuation ring with field of fractions $F$ and (sufficiently large) residue field $k$. We prove that there is a natural exact sequence $H_3($\mathrm{S}\mathrm{L}$_2(A),$\mathbb{Z}$[\frac{1}{2}]) \to H_3($\mathrm{S}\mathrm{L}$_2(F),$\mathbb{Z}$[\frac{1}{2}])\to $\mathcal{R}\mathcal{P}$_1(k)[\frac{1}{2}]\to 0$,where$$\mathcal{R}\mathcal{P}$_1(k)$ is the refined scissors congruence group of $k$. Let ${\mathrm{\Gamma }}_0({\mathfrak{m}}_A)$ denote the congruence subgroup consisting of matrices in ${\mathrm{S}\mathrm{L}}_2(A)$ whose lower off-diagonal entry lies in the maximal ideal ${\mathfrak{m}}_A$. We also prove that there is an exact sequence $0\to \overline{$\mathcal{P}$}(k)[\frac{1}{2}]\to H_2(\Gamma_0(\mathfrak{m}_A),$\mathbb{Z}$[\frac{1}{2}])\to H_2($\mathrm{S}\mathrm{L}$_2(A),$\mathbb{Z}$[\frac{1}{2}])\to I^2(k)[\frac{1}{2}]\to 0$, where $I^2(k)$ is the second power of the fundamental ideal of the Grothendieck-Witt ring $\mathrm{G}\mathrm{W}(k)$ and $\bar{\mathcal{P}}(k)$ is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) $\mathcal{P}(k)$ of $k$.