Characterization of class III graphs for the adjacent vertex distinguishing index by sum

For a graph $\Gamma$, let $tndi_{\sum}(\Gamma)$ be the minimum number of colors in a proper total coloring distinguishing adjacent vertices by sums. The graph is called $tndi_{\sum}$ class III when $tndi_{\sum}(\Gamma)=\Delta(\Gamma)+3$, where $\Delta(\Gamma)$ is its maximum degree. Class III characterization conjecture. A graph $\Gamma$ is $tndi_{\sum}$ class III if and only if

$$\Gamma=K_{2n+1}$$

for some $n\geq 1$. This proposes that the odd complete graphs are exactly the extremal graphs for the adjacent vertex distinguishing index by sum.