Uniform sign-change conjecture for lattice random-walk averages

Let $d$ be fixed. For a finitely supported function $\xi:{\mathbb Z}^d\to{\mathbb R}$, let $W_{2n}({\bf v},\xi)$ denote the associated lattice random-walk quantity at ${\bf v}$ and time $2n$. Uniform sign-change conjecture. For every such $\xi$ and every ${\bf v}\in{\mathbb Z}^d$, the function $W_{2n}({\bf v},\xi)$ has at most $K$ sign changes as a function of $n$, where $K$ depends on $\xi$ and $d$ but not on ${\bf v}$. The conjecture would control the alternating sums in the proof and reduce the need to bound the error terms; it is motivated by the expectation that oscillations in the random-walk kernel occur only modulo $2$.