Note: I have edited the post below in order to include sharper (conjectured) inequalities, using $|H_1 \cap H_2|$.
Let $[n] = \{1, \dots, n\}$ and let $\sim$ be an equivalence relation on $[n]$. Then $\sim$ induces a partition of $[n]$ into equivalence classes. Let $H$ denote the product of the groups of permutations corresponding to each of the equivalence classes induced by $\sim$.
For example, if $n = 7$ and the equivalence classes are $\{1, 2, 3\}$, $\{4, 5\}$ and $\{6, 7\}$, then $H$ is the product of the group of permutations of $\{1, 2, 3\}$, the group of permutations of $\{4, 5\}$ and the group of permutations of $\{6, 7\}$.
We will actually assume we have two equivalence relations on $[n]$, say $\sim_i$, for $i = 1, 2$ and, correspondingly, two groups of permutations, say $H_i$, for $i = 1, 2$.
Let $A$ be a complex $n$ by $n$ matrix. Define $$ f_{\sim_1, \sim_2}(A) = \sum_{\sigma_i \in H_i} \prod_{i = 1}^n A_{\sigma_1(i) \sigma_2(i)}.$$
I conjecture that, whenever $A$ is hermitian positive semidefinite, then $$ \operatorname{Re}(f_{\sim_1, \sim_2}(A)) \geq |H_1 \cap H_2| \prod_{i = 1}^n A_{ii}.$$
Note that if $H_1$ is trivial and $H_2$ is $S_n$, then the LHS is the permanent of $A$ and the desired inequality is then the famous Marcus's inequality.
So, if true, the conjecture above would generalize Marcus's inequality. I ran some numerical simulations in Python for various small values of $n$ (up to $n = 8$), with a few different pairs of equivalence classes and about 1000 random $A$s (hermitian positive semidefinite matrices) each time, and I haven't found any counterexample yet.
Is the above conjecture known? Does anyone know how to prove it, by any chance?
Edit 1: assuming it is true, does anyone know how to show nonnegativity of $\operatorname{Re}(f_{\sim_1, \sim_2}(A))$? That would be a nice step too!
Edit 2: it can be shown that the above conjecture is equivalent to the following. Let $v_i \in \mathbb{C}^n$, for $i = 1, \dots, n$ and form the following Kronecker product $$T = v_1 \otimes \dots \otimes v_n.$$ Let $$S_iT = \sum_{\sigma \in H_i} v_{\sigma^{-1}(1)} \otimes \dots \otimes v_{\sigma^{-1}(n)},$$ for $i = 1, 2$.
Then the conjecture above is equivalent to the following inequality $$ \operatorname{Re} \langle S_1 T, \, S_2 T \rangle \geq |H_1 \cap H_2| \prod_{a = 1}^n \lVert v_a \rVert^2, $$ where $\langle -, \, - \rangle$ is the hermitian inner product on $\bigotimes_{a = 1}^n \mathbb{C}^n$ induced by the standard hermitian inner product on $\mathbb{C}^n$.
