The branching conjecture for partition complexes

Let $\Lcal_n$ be the partition complex, let $\Pcal_n^\diamond$ denote the unreduced suspension of the partition complex associated to the partition lattice, and let $S^{\bar{\rho}_{n}}$ be the one-point compactification of the reduced standard representation of $\Sigma_n$ on $\mathbb{R}^{n-1}$. Regard $\Sigma_n$ as a subgroup of $U(n-1)$ via the permutation representation on the orthogonal complement of the diagonal in $\mathbb{C}^n$. Branching conjecture. There is a $U(n-1)$-equivariant homotopy equivalence

$$\Lcal_n \simeq U(n-1)_+ \wedge_{\Sigma_n}\left(\Pcal_n^\diamond \wedge S^{\bar{\rho}_{n}}\right).$$

This conjecture is motivated by orthogonal calculus and Goodwillie's homotopy calculus. The equivalence is known after taking suspension spectra and smashing with $EU(n)_+$, but the unstable equivariant homotopy equivalence remains conjectural.