Let $\mathcal{H}_N$ denote the cone of hermitian $N$ by $N$ positive semidefinite matrices and let $H \in \mathcal{H}_N$. We associate to $H$ the following complex numbers:
\begin{align*} c_1 &= H_{11} \\ c_2 &= c_1 H_{22} + c_1\{H_{1*} \mapsto H_{12}H_{2*}\} \\ c_3 &= c_2 H_{33} + c_2\{H_{*2} \mapsto H_{*3}H_{32}\} \\ & \vdots \\ c_n &= \left\{ \begin{array}{cl}c_{n-1} H_{nn} + c_{n-1}\{H_{n-1,*} \mapsto H_{n-1,n} H_{n*}\} & \text{if $n$ is even} \\ c_{n-1} H_{nn} + c_{n-1}\{H_{*,n-1} \mapsto H_{*n} H_{n,n-1} \} & \text{if $n$ is odd} \end{array} \right. \\ \end{align*}
for $2 \leq n \leq N$. By $c_{n-1}\{H_{n-1,*} \mapsto H_{n-1,n} H_{n*}\}$, I mean you take the expression for $c_{n-1}$ and you make the following substitutions: whenever you see $H_{n-1,*}$ (here $*$ could be any integer, though for this specific sequence, it would be an integer between $1$ and $n-1$) you replace it with $H_{n-1,n} H_{n*}$ where the $*$ is the same previous integer.
Thus, for example
\begin{align*} c_2 &= H_{11}H_{22} + H_{12} H_{21} \\ c_3 &= H_{11} H_{22} H_{33} + H_{12} H_{21} H_{33} + H_{11} H_{23} H_{32} + H_{13}H_{32} H_{21} \end{align*}
and so on.
I conjecture that $\Re(c_n) \geq H_{11} \cdots H_{nn}$, for $n = 1, \ldots N$. I know how to prove it for $n \leq 3$, but I am not yet sure how to prove it in general, assuming it is actually true. I do have the impression that a proof by induction may work. This would be fine for the first term on the RHS of $c_n$, using the induction hypothesis. But I get stuck at the second term (the one involving substitutions). Your help is kindly appreciated.