Non-holonomicity conjecture for the two-player winning probability

Let $p=q=1/2$, and let $W_{n,0,0}(x)$ be the probability generating function for the two-player pile game with boundary, with $n$ the initial pile size. Define the first player's winning probability by

$$\bar{w}(n):=W_{n,0,0}(1).$$

A sequence is holonomic if it satisfies a linear recurrence with polynomial coefficients. Non-holonomicity conjecture. The winning probability sequence $\bar{w}(n)$ is not holonomic: there are no specific $N$ and polynomials $p_0(n),p_1(n),\dots,p_N(n)$ such that

$$p_0(n)\bar{w}(n)+p_1(n)\bar{w}(n-1)+\dots+p_N(n)\bar{w}(n-N)=0.$$

The conjecture concerns the apparent absence of a linear recurrence for the two-player winning probabilities, in contrast with related pile games where the corresponding sequence is holonomic of small order. The evidence presented is computational and the conjecture remains open.