Sixth-moment conjecture for high-dimensional oriented percolation

Consider sufficiently spread-out oriented percolation in dimension $d>4$, with spread parameter $L$ sufficiently large, and let $\mathcal {T}_n$ denote the set of vertices connected to the origin at time $n$. Sixth-moment conjecture. There exists a constant $C_L$ such that, for every $n\in\mathbb{Z}_+$,

$$\sum_x |x|^6 \mathbb{P}(x\in\mathcal {T}_n)\le C_L n^3.$$

This bound would verify the remaining sixth-moment condition needed for the oriented-percolation analysis and hence support the predicted high-dimensional scaling results; it is conjectural in the source.