Poincaré embedding multiple disjunction conjecture

Let $N$ and $P$ be Poincaré spaces, and let $Q_1,ots,Q_r$ be Poincaré spaces embedded in $N$, with $E(P,N-Q_\bullet)$ denoting the $r$-cube of spaces of embeddings of $P$ into the complements $N-Q_S$. Assume the homotopy codimension hypotheses and set

$$\Sigma=\sum_{i=1}^{r}(n-q_i-2).$$

Poincaré embedding multiple disjunction conjecture. If the handle codimension restrictions are omitted and $r\geq1$, then the $r$-cube $E(P,N-Q_\bullet)$ is $(1-p+\Sigma)$-cartesian. This extends the proven multiple-disjunction estimate for Poincaré embeddings beyond the handle-codimension range and predicts the corresponding connectivity of the total homotopy fiber.