I have observed in literature that there is a conflicting naming convention for the superskewness. Almost exclusively this term is used in the fluid mechanics community in the study of turbulence. The conflict is as follows:

Some papers call the 5th normalized moment the superskewness,
where as other papers call the 6th normalized moment the superskewness.

It would seem that on face value the superskewness should be an odd moment due to the use of the word "skewness" referring to the 3rd normalized moment skewness. Following this idea, the 6th centralized moment should be the superkurtosis. Hyperskewness and hyperkurtosis should also follow from this.

Outside the fluid mechanics community there are a couple of references in the optics community that uniformly use the naming of the 5th normalized moment as the superskewness.

I have struggled to find significant references on this in the mathematical statistics community.

I was wondering if there was an authority from the mathematical statistics community on this topic in the maths help forum that could perhaps clear the correct naming convention for me (and others).