is a ‘yes’-instance.

Conversely, suppose that $I$ is a ‘yes’-instance of [4-Disjoint-Intermezzo]{}. Let ${{\mathcal O}}'$ be a [4-Disjoint-Intermezzo]{} ordering on the elements of $A$. To ease reading, let $$C=\bigcup_{1 \leq \ell \leq 4}C_\ell.$$ Modify ${{\mathcal O}}'$ as follows to obtain an ordering ${{\mathcal O}}$.

(1) Concatenate ${{\mathcal O}}'$ with the sequence $(d_1,d_2,\ldots,d_{|A'|})$.

(2) For each $c_r=(a_i,a_j,a_k)$ in $C$, do one of the following two depending on the order of $a_i$, $a_j$, and $a_k$ in ${{\mathcal O}}'$. If $a_i<a_j<a_k$ in ${{\mathcal O}}'$, then replace $a_i$ with $a_i,c_r^2$ and replace $a_k$ with $a_k,c_r^3,c_r^1,c_r^4$. Otherwise, if $a_j<a_k<a_i$, replace $a_k$ with $a_k,c_r^3$ and replace $a_i$ with $a_i,c_r^2,c_r^1,c_r^4$.

Since ${{\mathcal O}}'$ is a [4-Disjoint-Intermezzo]{} ordering with $a_i<a_j<a_k$ or $a_j<a_k<a_i$ for each $(a_i,a_j,a_k)\in C$, it follows from the construction of ${{\mathcal O}}$ from ${{\mathcal O}}'$ that ${{\mathcal O}}$ is an ordering on the elements in $A'\cup D$. It remains to show that ${{\mathcal O}}$ is a cherry-picking sequence for ${{\mathcal P}}_I$. First, consider a pendant subtree with leaf set $\{a_i,a_j,d_q\}$ in ${{\mathcal T}}_\ell$ and ${{\mathcal T}}_\ell'$ for some $1 \leq \ell \leq 4$. By construction, $(a_i,a_j)$ is a pair in $B_\ell$ and, so, we have $a_i<a_j$ in ${{\mathcal O}}'$ and $a_i<a_j<d_q$ in ${{\mathcal O}}$. Second, consider 