Conjecture on the total second-order derivative mass in first-passage percolation

Let $W_n$ be the edge set in the first-passage percolation model, let $f^\tau$ be the passage-time function in the torus model, and let $\partial_M f^\tau$ denote the second-order environment derivative indexed by a two-element subset $M\subseteq W_n$.

Second-order mass conjecture. There exists a constant $C$ independent of $N$ such that

$$\sum_{M\subseteq W_n,\,|M|=2}\|\partial_M f^\tau\|_2^2\leq C N.$$

This estimate would control the numerator in the second-order variance bound and is presented as an unproved conjecture in the paper.