Fixed-degree NP-completeness conjecture for non-synchronizing colorings

Let $G$ be a primitive $k$-out graph, meaning a finite directed graph in which every vertex has out-degree $k$ and whose associated transition structure is primitive. Let $\mathcal C(G)$ be the set of colorings of $G$, and define

$$\#\mathrm{NSC}(G):=\bigl|\{\chi\in\mathcal C(G):\chi(G)\text{ is non-synchronizing}\}\bigr|.$$

Fixed-degree counting conjecture. The counting problem

$$G\longmapsto\#\mathrm{NSC}(G)$$

is $\#P$-complete on the class of primitive $k$-out graphs, and remains $\#P$-complete for fixed out-degree $k$, already for $k=2$.

This is presented after a proof of the corresponding decision problem and concerns the exact number of non-synchronizing colorings. The source provides no resolution status for this counting claim.