Here are my first draft PR to pass the async_generator test case provided by @njsmith.
I have reset gi_running value to 1 after gen_send_ex() and _gen_throw() calls in async generators and clear it to 0 when they close.
This PR obviously requires more test cases regarding athrow() calls, but I submit at this stage to check if my approach is correct first.

https://bugs.python.org/issue32526

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Hm.. my changes makes too many existing tests (test_asyncgen) failing except the new test case. 😱

Async generators awaiting its inner awaitables should ALLOW send() on its returned async iterator object (as the current Python does now but gi_running becomse False) while should FORBID another asend() on itself from outside. This PR forbids both cases and thus breaks. 😞
It seems non-trivial to fix this situation... I need deeper understanding on how async-generator is implemented in CPython (coroutines and generators individually seems to be not very difficult but it looks like that async-generator mixes them up...).

I hope @1st1 and @njsmith would have some better ideas than me!
At least, my try will help me understand what's going on there even if somebody else fixes this issue.

I'll take a look tomorrow.