Non-D-finiteness conjecture for generating functions of colored set partitions

Let $C_n(r)$ denote the number of colored set partitions of $[n]$ with colors from a set of size $r$. A formal power series is D-finite if it satisfies a linear differential equation with polynomial coefficients. Non-D-finiteness conjecture. The generating function

$$\sum_{n\geq 0} C_n(r)x^n$$

is not D-finite for integers $r\geq 3$. This is motivated by computational evidence for $r=3$ and by analogous results and heuristics for uncolored noncrossing partitions; the conjecture asserts that no such differential equation exists for any number of colors at least three.