Non-differential-algebraicity conjecture for Av(4231, 4123, 4312)

Let $\operatorname{Av}(4231,4123,4312)$ be the permutation class avoiding $4231$, $4123$, and $4312$, and let $F(x)=\sum_{n\geq 0}a_nx^n$ be its ordinary generating function. Non-differential-algebraicity conjecture. The generating function for $\operatorname{Av}(4231,4123,4312)$ is not differentially algebraic. The source presents this as a candidate for a compact counterexample to the Noonan–Zeilberger conjecture; no proof is given there.