Scott sobriety of countable meet-continuous complete lattices

Determine whether the Scott space of every countable meet-continuous complete lattice is sober.

Background

Every countable frame has a sober Scott space, and every Artinian meet-continuous complete lattice is algebraic and therefore Scott sober. However, the paper does not establish sobriety for all countable meet-continuous complete lattices.

The question isolates whether countability and meet continuity alone suffice for Scott sobriety in the complete-lattice case. The paper notes that an affirmative solution to the preceding Scott-product question would resolve this sobriety question affirmatively, while any counterexample would have to be non-Artinian.

References

Let $L$ be a countable meet-continuous complete lattice. Must its Scott space $\Sigma L$ be sober?

— Scott topologies on products of countable complete Heyting algebras  (2609.18032 - Xu, 16 Sep 2026) in Question 2.2, Section 6 (Further questions)

Is it consistent with ZFC that $s>\aleph_1$?

— Scott topologies on products of countable complete Heyting algebras  (2609.18032 - Xu, 16 Sep 2026) in Question 2.5, Section 6 (Further questions)