Locality conjecture for dimension-distinguishing graph properties

Consider graph properties of the random nearest neighbor trees $G_n$ in dimension $d$. A graph property distinguishes the dimension if its asymptotic behavior depends sufficiently on $d$ to identify the dimension from the graph's combinatorial structure. A graph property is local if it is determined by bounded-radius neighborhoods in the Benjamini–Schramm sense.

Locality conjecture. A graph property distinguishes the dimension if and only if it is local.

The conjecture contrasts the paper's dimension-dependent local parameters with non-local quantities such as typical distance to the root and graph diameter, which show no first-order dependence on the dimension. Whether locality exactly characterizes dimension-distinguishing properties remains open.