You shouldn’t assume that because the Elamite empire lasted till only 2300 years ago and I m guessing many of them fled to S Iraq after that. Admixture easily picks up even 5000 year old ancestry
The other hint is since most of S Iraqis score decent S Indian it must be old and was past down to the present population as they multiplied. A stronger case for new can be made is when only less than 1% of a population has a certain admixture

