A negative Schur coefficient for products of two chains

We prove that the product of chains $\mathbf{m}\times\mathbf{n}$ is not Schur positive whenever $n\ge4$ and $m\ge3n-1$. Writing $m=n+k$, we exhibit a negative coefficient indexed by $(2n+k-1,2n+k-3,\ldots,k+5,k,k,4)$ and evaluate it explicitly as $(n-2)!$ times a polynomial of degree three in $k$. O