Barrett-Johnson inequalities for totally nonnegative matrices

Given a matrix $A$, let $A_{I,J}$ denote the submatrix of $A$ determined by rows $I$ and columns $J$. Fischer’s Inequalities state that for each $n\times n$ Hermitian positive semidefinite matrix $A$, and each subset $I$ of $\{1,\dots ,n\}$ and its complement $I^c$, we have $\det (A)\le \det (A_{I,I})\det (A_{I^c,I^c})$. Barrett and Johnson (Linear Multilinear Algebra 34, 1993) extended these to state inequalities for sums of products of principal minors whose orders are given by nonincreasing integer sequences $({\lambda }_1,\dots ,{\lambda }_r)$, $({\mu }_1,\dots ,{\mu }_s)$ summing to $n$. Specifically, if ${\lambda }_1+\cdots +{\lambda }_i\le {\mu }_1+\cdots +{\mu }_i$ for all $i$, then $${\lambda }_1!\cdots {\lambda }_r!\sum _{(I_1,\dots ,I_r)}\det (A_{I_1,I_1})\cdots \det (A_{I_r,I_r})\ge {\mu }_1!\cdots {\mu }_s!\sum _{(J_1,\dots ,J_s)}\det (A_{J_1,J_1})\cdots \det (A_{J_s,J_s}),$$ where sums are over sequences of disjoint subsets of $\{1,\dots ,n\}$ satisfying $\mathrm{\mid }{I}_k\mathrm{\mid }={\lambda }_k$, $\mathrm{\mid }{J}_k\mathrm{\mid }={\mu }_k$. We show that these inequalities hold for totally nonnegative matrices as well.