Arithmetic kei coloring average-order conjecture

Let $n\in\mathbb{N}$ be squarefree, let $\mathscr{K}$ be a finite kei, and define

$$\operatorname{col}_{\mathscr{K}}(n)=\left|\operatorname{Hom}_{\mathscr{K}\operatorname{ei}}^{\operatorname{cont}}(\mathord{\text{kei}}_n,\mathscr{K})\right|$$

to be the number of continuous $\mathscr{K}$-colorings of the arithmetic kei of $n$. For $X\geq 1$, let

$$\operatorname{Avg}_{\mathscr{K}}(X)=\frac{\sum_{1\leq n\leq X,\ n\ \mathrm{squarefree}}\operatorname{col}_{\mathscr{K}}(n)}{\sum_{1\leq n\leq X,\ n\ \mathrm{squarefree}}1}.$$

Arithmetic kei coloring average-order conjecture. Let $\mathscr{K}\in\mathscr{K}\operatorname{ei}$ be finite. Then there exists $a_{\mathscr{K}}>0$ such that

$$\operatorname{Avg}_{\mathscr{K}}(X)=a_{\mathscr{K}}\log^{\deg P_{\mathscr{K}}}(X)\cdot(1+o(1))$$

as $X\to\infty$, where $P_{\mathscr{K}}$ is the integer-valued polynomial governing the average number of $\mathscr{K}$-colorings of random braid closures. This is motivated by the polynomial average for random braid closures and the analogy between squarefree integers and braid closures; the asserted asymptotic for arithmetic kei colorings remains open.