I'm not sure if this answers your question, but it is possible that there are many $(r_1, \dots, r_n)$ that maximize that the dimension. For example, in the case $n=7$ and when the dimensions of the vector spaces are $(10,10,10,10,10,10,10)$, the ranks that maximize are $(7, 3, 4, 6, 2, 8)$, $(7, 3, 5, 5, 2, 8)$, $(7, 3, 5, 5, 3, 7)$, $(8, 2, 5, 5, 2, 8)$, $(8, 2, 5, 5, 3, 7)$, and $(8, 2, 6, 4, 3, 7)$. So there are several subvarieties of maximal dimension that do not contain each other.