Are all constant-sum games games of pure conflict, or only zero-sum games?

The title essentially says it all. I am asking since I came across the following statement: “Zero-sum games are also known as games of pure conflict”. This seemed to imply that zero-sum and pure conflict are synonyms, which confuses me since I figured that all constant-sum games should be games of pure conflict, by definition. So, are all constant-sum games games of pure conflict or are there some constant sum games that do not follow this rule?

• There are underlying assumptions here which raise issues in reality. The first is that the game is zero-sum or constant-sum in utility terms rather than some other measure. But there is experimental evidence that individuals show different behaviour when faced with large changes in total utility: if the possible outcomes are $+10$ or $-10$ people often try hard to reduce the risk of a $-10$ outcome, and make different decisions when faced with possible outcomes of say $+560$ or $+540$ Jun 24 '20 at 11:24