Johnson–Bapat interlacing conjecture for mixed determinant polynomials

Let $A$ and $B$ be Hermitian $n\times n$ matrices, and for an index set $\mathcal{S}$ let $A[\mathcal{S}]$ denote the corresponding principal submatrix. Define

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

where $\mathcal{S}'$ is the complement of $\mathcal{S}$. Two real-rooted univariate polynomials have interlacing zeros when their zeros can be ordered alternately; by convention, the zeros of the zero polynomial interlace those of every real-rooted polynomial.

Johnson–Bapat interlacing conjecture. If $A$ is positive semidefinite and $\eta(zA,-B)$ is not identically zero, then, for every $1\leq j\leq n$, the zeros of

$$\eta(zA[\{j\}'],-B[\{j\}'])$$

interlace those of $\eta(zA,-B)$.

This generalizes Cauchy–Poincaré interlacing for a Hermitian matrix and its principal submatrices. The paper proves the assertion, resolving the conjecture.