The twelve-neighbor common-neighbor conjecture for close-packed configurations

Let $Z\subset{\mathbb{R}}^3$ satisfy $|z'-z|\geq 1$ for all $\\{z,z'\\}\subset Z$. For each $z\in Z$, define its nearest-neighbor set by

N(z):=\left\\{z'\in Z:|z'-z|=1\right\\}.

Common-neighbor conjecture. If $z,z'\in Z$ satisfy \\#N(z)=\\#N(z')=12 and $z\in N(z')$, then

\\#(N(z)\cap N(z'))\geq 4.

This remains an open problem and could provide a route to eliminate the necessity of the three-body potential $V_3$ in the analysis of face-centered cubic crystallization.