Language Server version: 2020.9.5
I'd expect that the tooltip for the completion remains open as I'm typing the function name.
Tooltip intermittently hides as I'm typing.

I'm thinking this might be fixed via: https://github.com/microsoft/pyright/commit/90c8cdd72c5a6e7480c9ae982b02f809ba2c2d49
But we'll have to check.
Yes, I discovered this bug independently and fixed it. I'm confident it's the same issue.
We checked, and it's fixed.
This issue has been fixed in version 2020.9.6, which we've just released. You can find the changelog here: https://github.com/microsoft/pylance-release/blob/master/CHANGELOG.md#202096-23-september-2020