I have K2 2..7.1 on Joomla 3.6.5 on php 7.x and now for test 5.6.30 under Apache.
I was getting gateway timeouts when saving just one category which had extra fields. I did not see any errors so transferred the site to my local server and ran it locally. Now I do not get timeouts but to save just the category takes about 4 minutes. I understand why the hosted version was having the session timeout.
If I remove all the items in that category it saves normally, then adding the items back at 30 item chunks progressively increases the category save time. So it looks like it just takes a long time to save. No errors reported.
The example has 160 items each with 6 extra fields: 2 simple text fields, two drop down with 2 and three items and two multi-select with 17 and 18 items. Perhaps this is a lot for K2?
Additional information, even though the gateway times out it looks like the category changes made were saved so this suggests the delay is after the initial save.
I guess I can live with this because the customer is unlikely to save the category but I felt it was worth sharing??
Any help appreciated.

This indicates a setup issue in your server environment.
Check your error_log for any warnings/ errors. Are you using any 3rd party K2 extensions?

This indicates a setup issue in your server environment.
Check your error_log for any warnings/ errors. Are you using any 3rd party K2 extensions?