Lieb's permanent dominance conjecture for immanants

Let $A$ be a positive semidefinite Hermitian matrix, let $\lambda\vdash n$ be a partition, let $\chi^{\lambda}$ be the irreducible character of $\mathfrak{S}_n$ indexed by $\lambda$, and let $\operatorname{Imm}_{\chi^{\lambda}}(A)$ denote the corresponding $\lambda$-immanant. Write $\operatorname{per}(A)$ for the permanent of $A$. Lieb's conjecture. For every such $A$ and $\lambda$, the permanent dominates the $\lambda$-immanant:

$$\chi^{\lambda}(1)\operatorname{per}(A)\geq \operatorname{Imm}_{\chi^{\lambda}}(A).$$

The paper describes determining the maximal immanant as a longstanding open problem, and presents this conjecture as Lieb's proposed answer; no resolution is supplied in the given text.