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Your operad is a special case suboperad of the "surgery cylinder" operad described in arxiv 1009.5025 (more recent version available here). See Section 8 and figures therein. In the notation of that paper, your operad corresponds to case where all the manifolds $M_i$ and $N_i$ are $n$-cubes and all the homeomorphisms $f_i$ are the identity.

The surgery cylinder operad can be thought of as describing a sequence of (generalized) surgeries on an initial manifold $M_0$, yielding a final manifold $N_0$. At the $i$-th stage ($1\le i \le k$) we remove a codimension-0 submanifold $M_i$ replace it with $N_i$, where $m_i$ and $N_i$ have the same boundary. To get your overlapping $n$-cubes operad, let $M_0$ be the "big" $n$-cube and $M_i = N_i$ be the $i$-th little $n$-cube. In other words we have a sequence of pointless surgeries in which little $n$-cubes are removed and replaced with identical copies of themselves.
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I just now noticed this question.

Your operad is a special case of the "surgery cylinder" operad described in arxiv 1009.5025 (more recent version available here). See Section 8 and figures therein. In the notation of that paper, your operad corresponds to case where all the manifolds $M_i$ and $N_i$ are $n$-cubes and all the homeomorphisms $f_i$ are the identity.

The surgery cylinder operad can be thought of as describing a sequence of (generalized) surgeries on an initial manifold $M_0$, yielding a final manifold $N_0$. At the $i$-th stage ($1\le i \le k$) we remove a codimension-0 submanifold $M_i$ replace it with $N_i$, where $m_i$ and $N_i$ have the same boundary. To get your overlapping $n$-cubes operad, let $M_0$ be the "big" $n$-cube and $M_i = N_i$ be the $i$-th little $n$-cube. In other words we have a sequence of pointless surgeries in which little $n$-cubes are removed and replaced with identical copies of themselves.