The universal covering component conjecture for connected types

Let $A$ be a connected type, let $\tilde A$ denote its universal $n$-covering, and let $\Covering[n](A)$ denote the type of $n$-coverings of $A$. The connected component of $\tilde A$ in $\Covering[n](A)$ is the component equivalent to $\trunc{n+1}A$.

Universal covering component conjecture. The connected component of $\tilde A$ in $\Covering[n](A)$ is $\trunc{n+1}A$.

This conjecture is proposed as a means of constructing deloopings of higher groups from universal coverings. The supplied text does not indicate whether it has been proved or remains open.