The elementary generation conjecture for $SL_2(\mathbb Z[t,t^{-1}])$

Let $\mathbb Z[t,t^{-1}]$ be the Laurent polynomial ring in one variable over the integers, let $SL_2(\mathbb Z[t,t^{-1}])$ be its special linear group, and let $E_2(\mathbb Z[t,t^{-1}])$ denote the subgroup generated by elementary matrices. Elementary generation conjecture.

$$SL_2(\mathbb Z[t,t^{-1}]) = E_2(\mathbb Z[t,t^{-1}]).$$

The source presents this as a concrete conjecture motivated by numerical evidence; whether every element is generated by elementary matrices remains open in the paper.