Multiple disjunction conjecture for embeddings of several Poincaré spaces

Let $N$ be a Poincaré space and let $Q_1,\dots,Q_r$ be Poincaré spaces embedded in $N$. Write $E(Q_\bullet,N)$ for the $r$-cube of embedding spaces associated with the $Q_i$, and set

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

Multiple disjunction conjecture for Poincaré embeddings. If $r\geq2$, then the $r$-cube $E(Q_\bullet,N)$ is $(3-n+\Sigma)$-cartesian. This is one of the paper's conjectural extensions of multiple disjunction without handle codimension restrictions; its asserted cartesianness gives a connectivity estimate for simultaneous disjunction of the embedded subspaces.