grammatical number - What is the correct plural of "mantis"?

This question is related to the plural of "octopus" (yet another ancient Greek loanword animal):

What is the plural of "mantis"?

Oxford Dictionaries suggests "mantis" or "mantises".

Merriam Webster and Dictionary.com suggest "mantises" or "mantes".

This page from the Iowa State University Entomology Department suggests "mantids" (emphasis mine).

Praying mantids (preferred plural form of mantis) have never been
numerous in Iowa and historically they were only common in the far
southeastern corner of the state.

Personally, I use a mixture of "mantises" and "mantes" and find the usage of "mantids" strange, but what is the most etymologically correct / most recommended plural for the animal?