Paired coalition number conjecture for complete binary trees

Let $T(h)$ denote a complete binary tree with height $h$, and let $PC(T(h))$ denote its paired coalition number.

Complete binary tree conjecture.

$$PC(T(h))=\begin{cases}5 & \text{if } h=2,\\3 & \text{if } h \text{ is odd},\\0 & \text{otherwise.}\end{cases}$$

This conjecture gives the proposed paired coalition number for every complete binary tree. The preceding argument establishes the value for $h=4$; the general assertion remains unresolved in the supplied text.