Pate's two-index permanent inequality challenge

Let $A\in{\cal H}_n$, and let $A(i|j)$ denote the matrix obtained by deleting row $i$ and column $j$. Pate's inequality challenge.

$$a_{11}\mathop{\rm per} A(1|1)-a_{12}\mathop{\rm per} A(1|2)-a_{21}\mathop{\rm per} A(2|1)+a_{22}\mathop{\rm per} A(2|2)\leqslant2\mathop{\rm per} A.$$

The source presents this as a consequence of the now-refuted $k=1$ eigenvalue conjecture, but does not state its independent resolution.