Wilfian-formula conjecture for self-avoiding walks

Let $O$ be the origin in $\mathbb Z^2$, and let $a_n$ be the number of self-avoiding walks from $O$ to $O$ of length $n$. A Wilfian formula of type (W1) is an algorithm computing $a_n$ in time polynomial in $n$.

Self-avoiding-walk Wilfian conjecture. The sequence $a_n$ has no Wilfian formula of type (W1).

The conjecture concerns a natural walk-counting sequence in a graph whose growth makes direct dynamic programming ineffective. The supplied text gives no evidence of resolution.