Weiss tower obstruction conjecture for embedding immersions

Let $P$ and $N$ be manifolds of dimensions $p$ and $n$, respectively, and let $f\colon P\to N$ be an immersion equipped with a lift $f_{j-1}\in E_{j-1}(P,N)$. For $j\geq 2$, let $C_{j-1}$ be the relevant coefficient spectrum, let $E'_j(f)$ be the complement of the fat-diagonal pullback, and let $\xi=(j-1)\tau_N-j\tau_P$ be the induced virtual bundle. Weiss tower obstruction conjecture. There is an obstruction

$$\mu(f_{j-1})\in\pi_{j-1}\left(C_{j-1}\wedge_{h\Sigma_j}E'_j(f)^\xi\right)$$

that vanishes when $f_{j-1}$ lifts to $E_j(P,N)$. Conversely, if

$$(j+1)p+2j-1\leq jn\qquad\text{and}\qquad p\leq n-3,$$

then vanishing of $\mu(f_{j-1})$ implies that $f_{j-1}$ lifts to an embedding of $P$ in $N$. This conjecture seeks to extend the complete obstruction theory for deforming immersions to embeddings beyond the metastable range, using the Weiss embedding tower; the stated dimension inequalities are the range in which the converse is proposed.