Field-extension conjecture for spectrally arbitrary zero-nonzero patterns

Let $\mathbb{F}\subset\mathbb{K}$ be an extension of fields, and let $S$ be a zero-nonzero pattern that is spectrally arbitrary over $\mathbb{F}$. Field-extension conjecture. Then $S$ is spectrally arbitrary over $\mathbb{K}$. This conjecture, stated as Conjecture 6 in the cited work, is false: a block-diagonal $4\times4$ pattern provides a counterexample over an extension field. Whether the assertion remains false when $S$ is required to be irreducible is left open.