D-algebraicity conjecture for Catalan subsequences

Let $(C_n)_{n\in\mathbb{N}}$ be the Catalan numbers, and let $d\in\mathbb{N}\!\setminus\!\{0\}$. A sequence is D-algebraic of order $2$ and degree $2d$ if it satisfies a polynomial difference equation involving three consecutive terms, $C_{dn}$, $C_{d(n+1)}$, and $C_{d(n+2)}$, of total degree $2d$. D-algebraicity conjecture for Catalan subsequences. The sequence $(C_{dn})_{n\in\mathbb{N}}$ is D-algebraic of order $2$ and degree $2d$. The claim is motivated by computations for $d\in\{1,2,3,4,5\}$, while the source expects a proof may follow from a dynamical-system approach; it does not state that the general claim is proved.