ZFC determination of the least cardinality of Scott-non-sober frames

Determine whether ZFC proves that the least cardinality of a frame with a non-sober Scott space is ℵ₁, and whether ZFC proves the stronger assertion that the least cardinality of a spatial frame with a non-sober Scott space is ℵ₁.

Background

The paper defines two least cardinalities: one for arbitrary frames with non-sober Scott spaces and one for spatial frames with non-sober Scott spaces. It establishes the ZFC bounds ℵ₁ ≤ s ≤ s ≤ c and proves that both least cardinalities equal ℵ₁ under the Continuum Hypothesis.

Without an assumption on the continuum, the paper does not determine whether the lower bound ℵ₁ is attained. The spatial assertion is equivalent to requiring a spatial Scott-non-sober frame at every uncountable cardinality, because the corresponding cardinal spectrum is upward closed.