Positive-drift conjecture for higher-order environment derivatives

Fix $k\geq 2$. Let $S$ be a reasonable set of $k$ edges, and let $\partial_S f$ be its environment derivative. The conjecture concerns every positive value $m$ in the discrete range of this derivative.

Positive-drift conjecture. There exists $\theta_k\in(0,1)$ such that, for most reasonable $k$-edge sets $S$ and every positive $m$,

$$\mathbb P(\partial_S f=-m)\leq\theta_k\mathbb P(\partial_S f=m).$$

This one-sided comparison would imply that $\mathbb E[\partial_S f]$ is comparable to $\mathbb P(\partial_S f\neq 0)$ for the relevant sets. The notion of “most reasonable” sets is informal in the supplied text.