Roots of determinant of matrix with polynomial entries — a generalization

For $1 le i, j le k$, consider $rho_{ij}$ which are equal to either zero or one such that $rho_{ii}=1$ and $rho_{ij}=0$ if and only if $rho_{ji}=0$. How to find the zeros of the determinant of the following matrix?

$$begin{bmatrix}
g_1(x) & -rho_{12}f_1(x) & cdots & -rho_{1,k}f_1(x) \
-rho_{21}f_2(x) & g_2(x) & cdots & -rho_{2,k}f_2(x) \
vdots & vdots & ddots & vdots \
-rho_{k1}f_k(x) & -rho_{k2}f_k(x) & cdots &g_k(x)
end{bmatrix}$$

where $f_i$s are $g_i$s are complex polynomials.
When all the $rho_{ij}=1$, I have got the answer in MO already at Roots of determinant of matrix with polynomial entries. The present question is a more general case.

I am not sure whether there is any direct answer to this question. I am looking for some reference where similar kinds of problems ars discussed.