Geometric orbit-growth conjecture for Bulgarian Solitaire necklaces

Let $P$ be a primitive necklace, and let $\mathcal{O}_{P^k}$ denote the Bulgarian Solitaire orbit parametrized by $P^k$. For a necklace $P'$, say that $P$ and $P'$ are complementary when they are related by swapping black beads and white beads. Geometric orbit-growth conjecture. For any primitive necklace $P$ with $|P|\geq 3$, there is an integer $c_P$ such that, for every $k\geq 2$,

$$\left|\mathcal{O}_{P^k}\right|=(c_P)^{k-1}\left|\mathcal{O}_P\right|.$$

Moreover, complementary primitive necklaces have the same growth constant: if $P$ and $P'$ are related by swapping black beads and white beads, then $c_P=c_{P'}$. The conjecture predicts geometric growth of orbit sizes for powers of primitive necklaces; data for primitive necklaces of lengths $4$ and $5$ are cited in the source, but no proof or resolution is provided.