Johnson's real-rootedness conjecture for mixed determinant polynomials

Let $A$ and $B$ be Hermitian matrices of the same order. For subsets $\mathcal{S}$ of the index set, write $A[\mathcal{S}]$ for the corresponding principal submatrix, let $\mathcal{S}'$ be the complement of $\mathcal{S}$, and define

$$\eta(A,B)=\sum_{\mathcal{S}}\det(A[\mathcal{S}])\det(B[\mathcal{S}']).$$

Johnson's conjecture. If $A$ is positive semidefinite, then the polynomial $\eta(zA,-B)$ has all real zeros.

The conjecture extends the fact that $\eta(zI,-B)=\det(zI-B)$ for Hermitian $B$. The paper proves this assertion, so it is now resolved.