Alazemi–Andelić–Simić conjecture on eigenvalues of chain graphs

A chain graph is a bipartite graph whose neighborhoods in each color class form a chain under inclusion. For a graph $G$ and a vertex $v\in V(G)$, write $G-v$ for the vertex-deleted induced subgraph, and let $\operatorname{mult}(\lambda,G)$ denote the multiplicity of an adjacency eigenvalue $\lambda$ of $G$. A vertex $v$ is downer with respect to $\lambda$ when

$$\operatorname{mult}(\lambda,G-v)=\operatorname{mult}(\lambda,G)-1.$$

Alazemi–Andelić–Simić conjecture. In any chain graph, every vertex is downer with respect to every non-zero eigenvalue. Equivalently, for every chain graph $G$, every $v\in V(G)$, and every non-zero adjacency eigenvalue $\lambda$ of $G$, $G-v$ does not have $\lambda$ as an eigenvalue.

The conjecture concerns the interaction between vertex deletion and the non-zero adjacency spectrum of chain graphs. The paper notes that all non-zero eigenvalues of chain graphs are simple, so the question is whether deletion always reduces their multiplicity to zero; no resolution is given here.