Poincaré embedding-to-mapping conjecture

Let $N$ and $P$ be Poincaré spaces, and let $Q_1,\dots,Q_r$ be Poincaré spaces embedded in $N$. Let $\mathit{EF}(P,N-Q_\bullet)$ denote the $(r+1)$-cube obtained from the cube of Poincaré embedding spaces by mapping each embedding to its underlying map. Assume the homotopy codimension hypotheses and $r\geq1$. Poincaré embedding-to-mapping conjecture. The $(r+1)$-cube $\mathit{EF}(P,N-Q_\bullet)$ is $(n-2p-1)$-cartesian. This is a stronger comparison estimate than the corresponding embedding-disjunction statement and is intended to hold without handle codimension restrictions.