The edge-injective neighbour-sum-distinguishing weighting conjecture

Let $G$ be a nice graph, meaning a graph with no connected component isomorphic to $K_2$. An edge-weighting is edge-injective if it assigns distinct weights to all edges, and $\chi^{e,1}_\Sigma(G)$ is the smallest $k$ for which $G$ admits an edge-injective neighbour-sum-distinguishing $k$-edge-weighting. Edge-injective neighbour-sum-distinguishing weighting conjecture. For every nice graph $G$,

$$\chi^{e,1}_\Sigma(G)=|E(G)|.$$

Since edge-injectivity gives the lower bound $|E(G)|$, the conjecture asserts that every nice graph admits a bijective assignment of the weights $1,\ldots,|E(G)|$ whose incident-weight sums distinguish adjacent vertices. It is related to the 1-2-3 Conjecture and would imply progress on equitable neighbour-sum-distinguishing weightings; its status is not resolved in the supplied text.