The conjectural structure of the second homology of ${\rm SL}_2(\mathbb{Z}[1/n])$

Let $n=p_1\cdots p_l$, with $l>1$, be the prime decomposition of $n$ such that $1<r_{p_1}\leq\cdots\leq r_{p_l}$; if $r_{p_j}=r_{p_{j+1}}$ for some $j$, assume that $p_j<p_{j+1}$. Here $\mathbb{F}_{p_j}^\times$ denotes the multiplicative group of the finite field with $p_j$ elements, and $q_i$ is the prime denoted by $p_i$ in the corresponding case. The integer $n$ is assumed not to be divisible by any of $2,3,5,7,13$. The conjecture. The group $H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ is given by the following cases:

$$H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})\simeq\mathbb{Z}^{r_{p_1}}\oplus\bigoplus_{j=2}^l\mathbb{F}_{p_j}^\times$$

when $p_1\equiv11\pmod {12}$;

$$H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})\simeq\mathbb{Z}^{r_{p_1}}\oplus\begin{cases}\mathbb{Z}/2\oplus\bigoplus_{j=2}^l\mathbb{F}_{p_j}^\times & \text{if }p_j\equiv1\pmod {4}\text{ for all }2\leq j\leq l,\mathbb{Z}/4\oplus\mathbb{Z}/((q_i-1)/2)\oplus\bigoplus_{\underset{j\neq i}{j=2}}^l\mathbb{F}_{p_j}^\times & \text{if }p_i\equiv3\pmod {4}\text{ for some }2\leq i\leq l\end{cases}$$

when $p_1\equiv5\pmod {12}$;

$$H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})\simeq\mathbb{Z}^{r_{p_1}}\oplus\mathbb{Z}/3\oplus\bigoplus_{j=2}^l\mathbb{F}_{p_j}^\times$$

when $p_1\equiv7\pmod {12}$; and

$$H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})\simeq\mathbb{Z}^{r_{p_1}}\oplus\begin{cases}\mathbb{Z}/6\oplus\bigoplus_{j=2}^l\mathbb{F}_{p_j}^\times & \text{if }p_j\equiv1\pmod {4}\text{ for all }2\leq j\leq l,\mathbb{Z}/12\oplus\mathbb{Z}/((q_i-1)/2)\oplus\bigoplus_{\underset{j\neq i}{j=2}}^l\mathbb{F}_{p_j}^\times & \text{if }p_i\equiv3\pmod {4}\text{ for some }2\leq i\leq l\end{cases}$$

when $p_1\equiv1\pmod {12}$. The conjecture asserts that the upper bound for $r_n$ in the preceding theorem is attained. The authors strongly suspect this structure but state that they currently have no definitive proof or disproof; it concerns the second integral homology of the indicated $S$-arithmetic group.