When importing multiple nested templates using API, import fails with error

code: -32602
message: Invalid params.
data: Incorrect item key "foo.bar1" provided for trigger expression on "Template Child".

Import using browser is sucessful.

Steps to reproduce:

1. create new (parent) template
2. add new item to (parent) template
3. create new (child) template
4. link parent template to child template
5. add new item to (child) template
6. create trigger in child template using child and parent item(s)
7. export both templates (in same xml)
8. import xml using API

In (more complex) case import import using API fails with different error:

data: -32500
message: Application error.
data: Item "Item name" on "Template My Template": application "App" does not exist.