Can a Jordan block be similar to a sparser matrix?

Question

This answer gives an example of a JNF matrix (with two Jordan blocks) that is less sparse (has less 0 entries) than another matrix in its similarity class. Over an algebraically closed field (preferably $\mathbb C$), can a Jordan block $J$ be similar to a matrix with more $0$ entries than $J$?

Answer 1

In any field of characteristic $p>0$, the answer to your question is “yes”. E.g. the $p\times p$ companion matrix $C$ for the characteristic polynomial $(x-1)^p=x^p+(-1)^p$ is sparser than its Jordan form, the unipotent Jordan block $J=J_p(1)$.

In a field of characteristic zero, I think the answer is “no” but I haven’t found a proof yet.