Multiple disjunction conjecture for spaces of embeddings

Let $N$ be a smooth compact $n$-dimensional manifold and assume $s\geq 2$. For $1\leq i\leq s$, let $(Q_i,\partial_0Q_i,\partial_1Q_i)$ be a smooth compact manifold triad, with the manifolds $\partial_0Q_i$ disjointly embedded in $\partial N$. For $S\subset\underline s$, let $Q_S=\coprod_{i\in S}Q_i$ and let $E(Q_S,N)$ be the space of embeddings fixed on $\partial_0Q_S=\coprod_{i\in S}\partial_0Q_i$. Multiple disjunction conjecture. In this situation the $s$-dimensional cubical diagram formed by the spaces $\{E(Q_S,N)\}$ is $(3-n+\sum_i(n-q_i-2))$-cartesian. This is presented as a different form of the preceding multiple-disjunction statement; the result belongs to the homotopy-theoretic study of spaces of embeddings, and its asserted connectivity is established in the range covered by the cited Goodwillie–Klein work.