Non-algebraicity conjecture for rational-function composition generating functions

Let $m>2$, let $p(z)\in\mathbb N(z)$ be a rational function with $p(1)>1$, and let $D(z)$ denote the generating function under consideration for the corresponding restricted compositions.

Non-algebraicity conjecture. The generating function $D(z)$ is not algebraic.

The claim is known for all odd $m>2$, while the paper reports no general proof for even $m>2$.