The hafnian analogue of Marcus's inequality for positive semidefinite matrices

Let $p$ be a positive integer and let $A=(a_{i,j})$ be a positive semidefinite symmetric real $n\times n$ matrix. Form the $2pn\times 2pn$ block matrix whose $2p\times 2p$ blocks are all equal to $A$.

Hafnian inequality. The hafnian of this block matrix is at least

$$(2p-1)!!^n\prod_i a_{i,i}^p,$$

with equality if and only if $A$ has a zero row or is a diagonal matrix.

This is presented as the case $p=1$ obtained from the preceding hafnian inequality together with Marcus's inequality; the general assertion is the proposed extension to arbitrary positive integers $p$.