Non-D-finiteness conjecture for the generating function of $\mathcal{A}v(A_{5,1})$

Let $A_{5,1}$ be the permutation class under discussion, and let $\mathcal{A}v(A_{5,1})$ denote the class of permutations avoiding the patterns in $A_{5,1}$. Its generating function is the ordinary generating function counting permutations in $\mathcal{A}v(A_{5,1})$ by length. A formal power series is D-finite if it satisfies a non-trivial linear differential equation

$$p_k(x)f^{(k)}(x)+p_{k-1}(x)f^{(k-1)}(x)+\cdots+p_0(x)f(x)+q(x)=0,$$

where the coefficients $p_i(x)$ and $q(x)$ are polynomials.

Non-D-finiteness conjecture. The generating function for $\mathcal{A}v(A_{5,1})$ is not D-finite.

The conjecture is motivated by computational evidence: 642 initial terms did not fit the generating function to the tested rational, algebraic, D-finite, or differentially algebraic forms. The computations are not dispositive, and the conjecture remains open.