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The problem of decomposing the Kronecker product of $S_n$ characters is oneof the last major open problems in the ordinary representation theory of thesymmetric group $S_n$. Here we prove upper and lower polynomial bounds for themultiplicities of the Kronecker product $\chi^\lm\otimes\chi^\mu$, where forsome 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.

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