prompt-toolkit based apps (ipdb, ipython) don't work in tox 4 environment

I discovered that apps based on prompt-toolkit don't work in tox 4 environment properly, initial report in https://github.com/tox-dev/tox/issues/2936

The bug doesn't seem to have any effect on tests themselves, only interactive session used within tox 4.

MWE

# Assuming py311 env has prompt_toolkit installed
tox exec --skip-pkg-install -e py311 -- python3

Simple prompt from documentation is not displayed correctly:

from prompt_toolkit import prompt

text = prompt('Give me some input: ')  # Clears screen, but nothing can be seen only a cursor on first line and correct column.
print('You said: %s' % text)

It seems there is an issue with flushing the screen, but I'm not sure on which part. In docs, I haven't found anything that would help me resolve the problem.

Start by reproducing the MWE with tox exec --skip-pkg-install -e py311 -- python3 and the documented prompt_toolkit.prompt example. Trace the interactive prompt display and screen-flushing path affected by the tox 4 environment. Done means the prompt is visible and accepts input, with the entered text printed correctly.