Permanental inequalities and unit interval orders

Given an $n \times n$ matrix $A = (a_{i,j})$ and sets $I$, $J$ of $[1,n] := \{1,\dotsc,n\}$, 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