### Problem description
Ref #1210
### Implementation description
Call
[`SetFileTime()`](https://learn.microsoft.com/en-us/windows/win32/api/fileapi/nf-fileapi-setfiletime)
everytime `FileProvider::save()` is called.
### Additional things
I moved the call to `File::close()` from `FileProvider::open()` to
`FileProvider::close()` because `SetFileTime()` requires a file handler
as input, so I need `File::m_file` to be valid.
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