Block Lanczos successive-iteration interlacing conjecture

Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix and let $v\in\mathbb{R}^{n\times p}$ be a block vector. Let $s$ be the largest index such that the block Krylov subspace $\mathcal{K}_{s}(A,v)$ has full dimension. For $0<k<s$, let $T_k$ be the symmetric block tridiagonal matrix generated at the $k$th iteration of the block Lanczos algorithm applied to $A$ and $v$, with spectral decomposition whose Ritz values are $\theta_1^{(k)},\ldots,\theta_{kp}^{(k)}$. Block Lanczos interlacing conjecture. Each open interval

$$(\theta_i^{(k)},\theta_{i+p}^{(k)}),\qquad i=1,\ldots,(k-1)p,$$

contains at least one Ritz value of $T_j$ for every $j$ satisfying $k<j\leq s$. To the best of the authors' knowledge, no result was known that generalized the corresponding single-vector property to symmetric block tridiagonal matrices; the conjecture proposes precisely this generalization of interlacing across subsequent block Lanczos iterations.