Permanental inequalities and unit interval orders

Given an $n\times n$ matrix $A=(a_{i,j})$ and sets $I$, $J$ of $[1,n]:=\{1,\dots ,n\}$, let $A_{I,J}=(a_{i,j}{)}_{i\in I,j\in J}$ denote the $(I,J)$-submatrix of $A$. We consider a class of $0$-$1$ totally nonnegative (TNN) matrices arising as antiadjacency matrices of unit interval orders, and show that for each matrix $A$ in this class, the inequalities \begin{equation} \operatorname{perm}(A_{[1,h],[1,h]}) \operatorname{perm}(A_{[h+1,n],[h+1,n]}) ;\geq; \operatorname{perm}(A_{[1,n] \cap 2\mathbb Z, [1,n] \cap 2\mathbb Z}) \operatorname{perm}(A_{[1,n] \smallsetminus 2\mathbb Z, [1,n] \smallsetminus 2\mathbb Z}) \label{eq:absmain} \end{equation} hold for $h=1,\dots ,n-1$. Let ${\mathfrak{A}}_n$ be the Young subgroup ${\mathfrak{S}}_{1,\dots ,\lfloor \frac{n}{2}\rfloor }\times {\mathfrak{S}}_{\lfloor \frac{n}{2}\rfloor +1,\dots ,n}$, and ${\mathfrak{B}}_n$ be the parity-alternating permutations \{w = w_1\cdots w_n \in \mathfrak{S}_n \ | \ \text{i$and$w_ihave the same parity} \}. We find a bijective map $f_n:{\mathfrak{A}}_n\to {\mathfrak{B}}_n$ such that $w\le f_n(w)$ under the Bruhat order for all $n\ge 4$. The existence of such a bijective map $f_n$ ensures that inequalities are satisfied for all TNN matrices $A$ when $h=\lfloor \frac{n}{2}\rfloor$. We also conjecture the inequalities to hold for all totally nonnegative matrices and all $h=1,\dots ,n-1$