Generic double multiplicity conjecture for symmetric singular matrix pencils

Let $A-\lambda B$ be a real symmetric singular $n\times n$ pencil with normal rank $n-k$, and let $D_A-\lambda D_B$ be a real symmetric regular $k\times k$ pencil. For a matrix $U\in\mathbb R^{n,k}$ and a nonzero scalar $\tau\in\mathbb R$, consider the rank-$k$ perturbation

$$A-\lambda B+\tau U(D_A-\lambda D_B)U^\top.$$

Generic double multiplicity conjecture. There exists a generic set $\Omega\subseteq\mathbb R^{n,k}$ such that, for every $U\in\Omega$ and every $\tau\in\mathbb R\setminus\{0\}$, the spectrum of the perturbed pencil consists of the eigenvalues of $A-\lambda B$ (true eigenvalues), the eigenvalues of $D_A-\lambda D_B$ (prescribed eigenvalues), and random eigenvalues, all of which have algebraic multiplicity precisely two. This conjecture extends the rank-one result for pencils with one minimal index; the observed real-symmetric examples exhibit random eigenvalues in double pairs.