Pham's duality conjecture for Bulgarian Solitaire orbit-growth constants

Let $P$ be a primitive necklace, let $P^*$ be its dual necklace, and let $c_P$ and $c_{P^*}$ denote the integers whose existence is asserted by the orbit-size growth conjecture when they exist.

Pham's duality conjecture. For any primitive necklace $P$ such that $c_P$ and $c_{P^*}$ both exist,

$$c_P=c_{P^*}.$$

This predicts that dual primitive necklaces have the same orbit-growth constant. The supplied text gives special examples but no general proof or disproof, so the conjecture remains open.