The D-antimagic neighborhood conjecture

Let $G$ be a graph, let $D\subseteq\{0,1,2,\dots,\operatorname{diam}(G)\}$ be a non-empty distance set, and define the $D$-neighborhood of a vertex $u$ by

$$N_D(u)=\{v\mid d(v,u)\in D\}.$$

A bijection $f:V(G)\rightarrow\{0,1,2,\dots,\operatorname{diam}(G)\}$ is a $D$-antimagic labeling when the $D$-weights

$$\omega_D(u)=\sum_{v\in N_D(u)}f(v)$$

are distinct for every vertex $u$; in that case $G$ is $D$-antimagic. The D-antimagic neighborhood conjecture. A graph $G$ is $D$-antimagic if and only if every vertex in $G$ has a distinct $D$-neighborhood. This generalizes the distance-antimagic claim when $D=\{1\}$; the supplied text does not establish the conjecture beyond presenting it as a proposed generalization.