a pendant subtree with leaf set $\{a_i,a_j,a_k,c_r^1,c_r^2,c_r^3,c_r^4,d_q\}$ in ${{\mathcal T}}_\ell$ and ${{\mathcal T}}_\ell'$ for some $1 \leq \ell \leq 4$. By construction, $(a_i,a_j,a_k)$ is a triple in $C_\ell$ and, so, we have either $a_i<a_j<a_k$ in ${{\mathcal O}}'$ and $$a_i<c_r^2<a_j<a_k<c_r^3<c_r^1<c_r^4<d_q$$ in ${{\mathcal O}}$, or $a_j<a_k<a_i$ in ${{\mathcal O}}'$ and $$a_j<a_k<c_r^3<a_i<c_r^2<c_r^1<c_r^4<d_q$$ in ${{\mathcal O}}$. Third, consider a pendant subtree with leaf set $\{a_i,d_q\}$ in ${{\mathcal T}}_\ell$ and ${{\mathcal T}}_\ell'$ for some $1 \leq \ell \leq 4$. By construction, we have $a_i<d_q$ in ${{\mathcal O}}$. Fourth, if $(a_i,a_j,a_k)\in C$, then, as $I$ has a [4-Disjoint-Intermezzo]{} ordering, there does not exist a pair $(a_k,a_j)$ in $B_\ell$ for some $1 \leq \ell \leq 4$. Lastly, observe that $(d_1,d_2,\ldots,d_{|A'|})$ is a suffix of ${{\mathcal O}}$ and that, for any two trees, say ${{\mathcal S}}$ and ${{\mathcal S}}'$ in ${{\mathcal P}}_I$, we have that ${{\mathcal S}}|D$ and ${{\mathcal S}}'|D$ are isomorphic. Since ${{\mathcal O}}'$ is a [4-Disjoint-Intermezzo]{} ordering, it is now straightforward to check that ${{\mathcal O}}$ is a cherry-picking sequence of ${{\mathcal P}}_I$. This establishes the proof of the claim and, thereby, the theorem.

The next corollary shows that [CPS-Existence]{} is not only NP-complete for a collection of eight rooted binary phylogenetic trees on the same 