The non-delooping conjecture for mapping spaces of classifying-space skeleta

Let $G$ be a non-commutative compact connected Lie group, let $B_nG$ denote the $n$-th stage of the classifying-space construction, and let $\iota_n\colon B_nG\to BG$ be the canonical map. For $1\le n<\infty$, consider the mapping space $\operatorname{Map}(B_nG,BG;\iota_n)$ in the component of $\iota_n$. Non-delooping conjecture. The space

$$\operatorname{Map}(B_nG,BG;\iota_n)$$

is never delooped. This conjecture asserts non-extendability of the evaluation fiber sequence discussed above; the preceding examples show both a non-deloopable mapping space in a related case and a contrasting torus case where the relevant maps are weak equivalences.