I asked this question on Math.SE here, but did not get a lot of attention.

I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over a field of positive characteristic $p$. It is enough to consider the case of single Jordan block.

So let $u$ and $v$ be unipotent Jordan blocks, of sizes $n \times n$ and $m \times m$ respectively.

In characteristic $0$, if $n \geq m$ then $u \otimes v$ has Jordan blocks of sizes $n + m - 1, n + m - 3, \cdots, n - m + 1$. This formula no longer works in positive characteristic. But apparently it is valid if $p \geq m+n$.

What is known about the Jordan decomposition of the tensor product in general? Is this open? I would like some useful description of the decomposition in terms of $n, m$ and $p$.