Kronecker multiplicities in the $(k,\ell)$ hook are polynomially bounded

The problem of decomposing the Kronecker product of $S_n$ characters is one of the last major open problems in the ordinary representation theory of the symmetric group $S_n$. Here we prove upper and lower polynomial bounds for the multiplicities of the Kronecker product $\chi^\lm\otimes\chi^\mu$, where for some fixed $k$ and $\ell$ both partitions $\lm$ and $\mu$ are in the $(k,\ell)$ hook, $\lm$ and $\mu$ are partitions of $n$, and $n$ goes to infinity.