Partition-dependence conjecture for divisor and orbit polynomial leading coefficients

Let $X$ be a $Z_n$ curve described by the family in the source. For each positive integer $c$, let $x_c$ be the number of indices $i$ with $c_i=c$, and for each positive integer $d$, let $y_d$ be the number of indices $i$ with $d_i=d$:

$$x_c=\left|\{i\mid c_i=c\}\right|,\qquad y_d=\left|\{i\mid d_i=d\}\right|.$$

Thus $p=\sum_c x_c$ and $q=\sum_d y_d$ determine partitions of $p$ and $q$, respectively. Partition-dependence conjecture. The leading coefficients of the two polynomials in the polynomial-growth conjecture depend only on these partitions of $p$ and $q$. This is a refinement of the preceding conjecture: the source reports it as a pattern suggested by the numerical examples, without proving the asserted dependence.