On the nonnegativity of monomial immanants for hook partitions

Let $A=(a_{ij})$ be an $n\times n$ real matrix and let $\lambda$ be a partition of $n$. Let ${\phi }^{\lambda }$ be the class function dual to the Young permutation character, and let $${\phi }^{\lambda }[A]=\sum _{\sigma \in {\mathfrak{S}}_n}{\phi }^{\lambda }(\sigma )\prod _{i=1}^n{a}_{i\sigma (i)}$$ be the corresponding monomial immanant. Stembridge [Canad. J. Math. 44 (1992), pp. 1079-1099] posed the following open problem: If all minors of $A$ of order at most $r$ are nonnegative, and the partition $\lambda$ has length at most $r$, is it true that ${\phi }^{\lambda }[A]\ge 0$? This paper solves the problem for all hook partitions.