Since for an odd integer $n$, a complete graph on $n$ vertices is list-edge-$n$ choosable, and the total chromatic number is $n$, it is easy to see that the list total chromatic number is bounded above by $n+2$ by a greedy algorithm.
My question is, can the bound be reduced further to obtain, say the that the list total chromatic number is $n+1$, or the optimal $n$? I was thinking along the lines of the list total chromatic number can be bounded above by $\text{max(list chromatic index, total chromatic number)}+1$ for any graph. Any counterexamples to this fact? Thanks beforehand.
