Classification of homogeneous functors by classifying spaces of self-equivalences

Let $\mathcal{M}$ and $A$ be as in the preceding conjecture. For a manifold $M$, let $\mathcal{F}_{kA}(\mathcal{O}(M);\mathcal{M})/we$ denote the weak-equivalence classes of degree-$k$ homogeneous functors with value equivalent to $A$, and let $F_k(M)$ denote the configuration space of $k$ distinct points in $M$. Assume additionally, when $k\geq2$, that $\mathcal{M}$ has a zero object.

Classification conjecture. There are bijections

$$\mathcal{F}_{1A}(\mathcal{O}(M);\mathcal{M})/we\cong [M,\emph{Bhaut} A]$$

when $k=1$, and

$$\mathcal{F}_{kA}(\mathcal{O}(M);\mathcal{M})/we\cong [F_k(M),\emph{Bhaut} A]$$

when $k\geq2$.

This conjecture would classify homogeneous functors by homotopy classes of maps from the manifold, or from its unordered configuration space in higher degree, into the classifying space of self-weak equivalences of $A$. It depends on the preceding conjectural identification of $\widehat{A}$ with $\emph{Bhaut} A$.