Sublinear variance conjecture for direct products of infinite finitely generated groups

Let $G$ be the direct product of two infinite finitely generated groups. For a Cayley graph of $G$, let $d_{\omega}$ be the first-passage percolation metric and let $\bar d=\mathbb{E}d_{\omega}$. The variance estimate referred to as

concerns the fluctuations of $d_{\omega}$ around its mean. **Sublinear variance conjecture.** The estimate

is satisfied for all Cayley graphs of $G$.

This is proposed as a modest form of the expected connection between sublinear variance and the absence of cut points in asymptotic cones. The source does not state a resolution of the conjecture.