Class II characterization for regular graphs under adjacent vertex distinguishing total coloring by sum

Let $\Gamma$ be a $k$-regular graph. The graph is called $tndi_{\sum}$ class II when its adjacent vertex distinguishing index by sum attains the class-II value. Class II characterization conjecture. A $k$-regular graph $\Gamma$ is $tndi_{\sum}$ class II if and only if $\Gamma$ has a proper vertex $k+2$ coloring. The conjecture relates the class-II condition for regular graphs to ordinary proper vertex colorability with $k+2$ colors.