Multiple disjunction conjecture for spaces of Poincare embeddings

Let $N$ be a smooth compact $n$-dimensional manifold. Let $(Q_1,\dots,Q_r)$, with $r\geq 1$, be pairwise disjoint compact submanifolds of $N$ transverse to $\partial N$, and let $q_i$ be the handle dimension of $Q_i$ relative to $\partial_0Q_i=Q_i\cap\partial N$. Let $(P,\partial_0P,\partial_1P)$ be a compact manifold triad, let $p$ be the handle dimension of $P$ relative to $\partial_0P$, and suppose that an embedding $e_0:\partial_0P\rightarrow\partial N$ is given, disjoint from $\partial_0Q_i$ for all $i$. For $S\subset\underline r=\{1,\dots,r\}$, set $Q_S=\bigcup_{i\in S}Q_i$. Multiple disjunction conjecture. In this situation the $r$-dimensional cubical diagram formed by the spaces $\{E(P,N-Q_S)\}$ is $(1-p+\sum_i(n-q_i-2))$-cartesian. In the cited work, this is proved when $n-p\geq 3$ and $n-q_i\geq 3$; the conjecture concerns the remaining cases and expresses the expected multiple-disjunction connectivity for spaces of embeddings.