Bača–Jendrol'–Miller–Ryan's total edge irregularity strength conjecture

Let $G$ be a graph, let $\Delta(G)$ be its maximum degree, and let $|E(G)|$ denote its number of edges. The total edge irregularity strength $\operatorname{s}'_\Sigma{}^t(G)$ is the least $k$ for which a total $k$-weighting gives distinct values to all edges when each edge value is the sum of the weights on its endpoints and on the edge itself.

Bača–Jendrol'–Miller–Ryan's conjecture. If $G\not\cong K_5$, then

$$\operatorname{s}'_\Sigma{}^t(G)=\max\left\{\left\lceil\frac{\Delta(G)+1}{2}\right\rceil,\left\lceil\frac{|E(G)|+2}{3}\right\rceil\right\}.$$

The source reports substantial partial verification, including sufficiently dense graphs and several sparse classes, but not a complete proof.