section, we show that the decision problem [CPS-Existence]{} is NP-complete for any collection of rooted binary phylogenetic trees on the same leaf set that consists of a constant number $m$ of trees with $m\geq 8$. To establish the result, we use a reduction from [4-Disjoint-Intermezzo]{}.

![A caterpillar on $|A'|$ leaves and with cherry $\{D_1,D_2\}$.[]{data-label="fig:caterpillar"}](CaterpillarSkeleton){width=".35\textwidth"}

Let $I$ be an instance of [4-Disjoint-Intermezzo]{}. Using the same notation as in the definition of [$N$-Disjoint-Intermezzo]{}, let $$A'=A\cup\left\{c_r^1,c_r^2,c_r^3,c_r^4\mid c_r\in\bigcup_{1 \leq \ell \leq 4} C_\ell\right\},$$ and let $D=\{d_1,d_2,\ldots,d_{|A'|}\}$. For each $\ell\in\{1,2,3,4\}$, we next construct two rooted binary phylogenetic trees. Let $A_\ell$ be the subset of $A'$ that precisely contains each element of $A'$ that is neither contained in an element of $B_\ell$ nor contained in an element of $$C_\ell\cup \{c_r^1,c_r^2,\ldots,c_r^4 \mid c_r\in C_\ell\}.$$ Furthermore, let ${{\mathcal S}}_\ell$ and ${{\mathcal S}}_\ell'$ both be the caterpillar shown in Figure \[fig:caterpillar\]. Setting $q=1$, let ${{\mathcal T}}_\ell$ and ${{\mathcal T}}_\ell'$ be the two rooted binary phylogenetic trees obtained from ${{\mathcal S}}_\ell$ and ${{\mathcal S}}_\ell'$ that result from the following four-step process.

(i) For each $(a_i,a_j)\in B_\ell$ in turn, replace the leaf $D_q$ in ${{\mathcal S}}_\ell$ (resp. ${{\mathcal S}}_\ell'$) with the 3-taxon tree on the top left (resp. bottom left) in 