Let $X$ be a locally spectral topological space (i.e. a space admitting an open cover by spectral spaces). Does there necessarily exist a quasi-compact locally spectral space $Y$ and an injective continuous open map $X\rightarrow Y$? What if we require $Y$ to be spectral?
I still do not know the answer to the first question, but it turns out the second question was pretty idiotic.
Take any topological space that has a cover by two spectral open subspaces whose intersection is not quasi-compact (e.g. the underlying space of the infinite-dimensional affine space with doubled origin over the integers).
Notice that an injective continuous open map is an open topological embedding, and that in an open subspace of a spectral space the intersection of any two quasi-compact opens is quasi-compact. Therefore, the required map can not exist.