The downer-vertex conjecture for chain graphs

A chain graph is a bipartite graph whose neighborhoods in each color class form a chain under inclusion. For a graph $G$, a vertex is downer with respect to an eigenvalue $\lambda$ if deleting that vertex decreases the multiplicity of $\lambda$.

Downer-vertex conjecture. In any chain graph, every vertex is downer with respect to every non-zero eigenvalue.

Non-zero eigenvalues of chain graphs are simple, so chain graphs have no Parter vertices for non-zero eigenvalues. The conjecture asserts that neutral vertices cannot occur for such eigenvalues.