I don't think such a test suite exists because there is no standardized definition for Read, Write, and Seek. For example, some Read+Write+Seek implementations can have their data modified behind their back (e.g. std::fs::File), while this is statically prevented in other implementations (e.g. std::fs::Cursor).
I think the best example you'll find of a "well behaved" implementation are the std::io::Cursor tests.
I created my own version of these tests and ran into something unexpected; the assertion on the result of write_vectored fails because only 2 bytes are written when 3 are expected. & doesn't get written for some reason.
Since write_vectored is a provided method, I'm not sure where to start.
Doing the equivalent with write appears to work, and IoSlice::new(&[8, 9, 10]) also works.