Equal orbit-growth constants imply equal generating-function denominators

Let $P_1$ and $P_2$ be primitive necklaces, and let $c_{P_1}$ and $c_{P_2}$ be their orbit-growth constants when these constants exist. Let $H_{P_i}(x)$ denote the limiting level size generating function associated with $P_i$.

Equal-constant denominator conjecture. For any two primitive necklaces $P_1$ and $P_2$, if $c_{P_1}=c_{P_2}$, then $H_{P_1}(x)$ and $H_{P_2}(x)$ have the same denominator.

The conjecture is motivated by computed examples in which equal orbit-growth constants accompany equal denominators, while the paper does not provide a general proof or disproof. It therefore remains open.