Distance-two antimagic labelling conjecture

Let $G$ be a graph with $m$ edges and no isolated edge. For each vertex $v$, let $S_v$ be the sum of the labels on edges incident with $v$. A distance-two antimagic labelling uses every label from $[m]=\{1,\ldots,m\}$ exactly once and requires the vertex sums to distinguish vertices at graph distance at most two. Distance-two antimagic labelling conjecture. There is a labelling of $G$ using every label from $[m]$ exactly once such that

$$S_v\ne S_w$$

for every pair of vertices $v,w$ with $d(v,w)\le 2$. This is proposed as a natural next step toward the full antimagic labelling conjecture; its status is left open in the paper.