On the existence of special elements in odd $K$-theory groups

Let $k$ be an imaginary quadratic number field, and $F\mathrm{/}k$ a finite abelian extension of Galois group $G$. We investigate the relationship between the conjectural special elements introduced in \cite{Burns-DeJeu-Gangl} and ETNC in the semi-simple case. This provides a partial proof of the conjecture for $F\mathrm{/}k$ under certain conditions.