The Unreasonable Slightness Conjecture for elementary matrix groups

Let $\dim=3$, $4$, or $5$. Let $\mathcal{A}$ be either an imaginary quadratic field when $\dim=3$, a rational definite quaternion algebra equipped with an orthogonal involution $\ddagger$ when $\dim=4$, or a rational definite quaternion algebra when $\dim=5$. Let $\mathcal{O}$ be a maximal order (or $\ddagger$-order) of $\mathcal{A}$. Let $\Gamma$ be $SL(2,\mathcal{O})$ when $\dim=3,5$ and $SL^\ddagger(2,\mathcal{O})$ when $\dim=4$. Let $\mathcal{E}$ be the subgroup generated by upper and lower triangular matrices. The amended Unreasonable Slightness Conjecture. Exactly one of the following holds:

1. $\mathcal{O}$ is norm-Euclidean (or norm $\ddagger$-Euclidean when $\dim=4$) and $\Gamma=\mathcal{E}$.
2. $\mathcal{O}$ is not Euclidean (or $\ddagger$-Euclidean when $\dim=4$) and $\mathcal{E}$ is an infinite-index, non-normal subgroup of $\Gamma$.

This conjecture predicts a sharp dichotomy between Euclidean orders, for which elementary matrices generate the relevant arithmetic group, and non-Euclidean orders, for which they generate a small non-normal subgroup. Its status is not established by the supplied source context.