Show that a certain ratio of diagonal entries dominates a certain ratio of singular values.

Question

Let $D=[d_{ij}]_{i,j=1,\ldots,4}$ be a $4 \times 4$ “density matrix”, that is, a Hermitian (possibly symmetric) positive definite matrix having trace 1—that is, the (nonnegative) diagonal entries sum to 1. Let $D_1$ and $D_2$ be the two diagonal $2 \times 2$ blocks of $D$. Further, $p= \sqrt{d_{11} d_{44}/(d_{22} d_{33})}$ and $q$ equal the ratio of the two singular values of $D_2^{1/2} D_1^{-1/2}$. Show that $\min (p,1/p) \geq \min (q,1/q)$.