Kronecker multiplicities in the $(k,\mathrm{\ell })$ 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 ${\chi }^{\text{lm}}\otimes {\chi }^{\mu }$, where for some fixed $k$ and $\mathrm{\ell }$ both partitions $\text{lm}$ and $\mu$ are in the $(k,\mathrm{\ell })$ hook, $\text{lm}$ and $\mu$ are partitions of $n$, and $n$ goes to infinity.