  The mapping spaces of the $r$-truncated versions of the $E_n$ operads appear
as the $r$-th stage of the Taylor tower for long embedding spaces. It has been
shown that their rational homotopy groups can be expressed through graph
homology in the limit $r\to \infty$. For finite $r$ only a part of the graph
homology appears in the homotopy groups of the Taylor tower at this stage,
possibly along with some additional unstable homotopy groups. In this paper we
study the convergence properties in some more detail. In particular, we provide
bounds for the stage $r$ at which the various graph homology classes start
appearing, and we provide degree bounds for the unstable (i.e., vanishing as
$r\to \infty$) terms in the homotopy groups of the tower.
