F-test Stepwise regression

I have a question about F-test Stepwise regression.
https://en.wikipedia.org/wiki/F-test
Regression problems

When doing F-test stepwise regression,
F statistic is calculated as below. The right side follows F-distribution if (RSS1 - RSS2) and (RSS2) are independent each other, because independent random variables that follow chi-square distribution have the reproductive property, and both the denominator and the numerator follow chi-square distribution.
However, if they are not independent, it looks the right side may not always follow F-distribution mathematically strictly.
So, in case of the latter case, it seems F-statistic sometimes shows inappropriate result against our expectation, strictly speaking.
I tried to find whether they are always independent, but I couldn't find finally.
Is there any mathematically strict proof for this?
Or is there any approximation, that is, even if they are not always independent, this test is useful in almost cases, so some strict arguments are ignored?
If someone knows the answer, please write, thank you.