Uniqueness of higher deloopings in the Weiss tower

Let $V$ be a real inner-product space, let $QBE_k^{(\mathbb R\oplus V)\otimes_{\mathbb R}\rho_k}$ denote the source spaces in the Weiss tower, and let $\operatorname{hoNat}$ denote the homotopically meaningful space of natural transformations. Higher-delooping uniqueness conjecture. For all $r\geq 1$ and any $d>1$, the $(d-1)$-fold looping map

$$\operatorname{hoNat}\left[\Omega QBE_k^{(\mathbb R\oplus V)\otimes_{\mathbb R}\rho_k},\Omega QBE_{k+r}^{(\mathbb R\oplus V)\otimes_{\mathbb R}\rho_{k+r}}\right] \longrightarrow \operatorname{hoNat}\left[\Omega^dQBE_k^{(\mathbb R\oplus V)\otimes_{\mathbb R}\rho_k},\Omega^dQBE_{k+r}^{(\mathbb R\oplus V)\otimes_{\mathbb R}\rho_{k+r}}\right]$$

is bijective, and in particular one-to-one, on $\pi_0$. This is proposed as a generalization of the cited uniqueness theorem for $r=1$; in the case $r=2$, it would imply the nullity of the composite of two successive deloopings. The supplied text gives no resolution for general $r$.