Higher-order derivative probability-ratio conjecture in the torus model

Fix $k\geq 3$. Let $S$ be a sufficiently general $k$-element set of edges and let $S'\subset S$ have cardinality $k-1$. Write $\partial_S f$ and $\partial_{S'} f$ for the corresponding environment derivatives.

Derivative probability-ratio conjecture. There exists a constant $\theta\in(0,d/2)$ such that

$$n^\theta\mathbb P(\partial_S f\neq 0)\approx\mathbb P(\partial_{S'} f\neq 0).$$

This conjecture is identified as the harder estimate needed, for at least one $k\geq 3$, to improve variance bounds in the torus model. The meanings of “sufficiently general” and the comparison symbol $\approx$ are not defined in the supplied text.