Universality versus total universality

Determine whether the universality property of a Borel relation already implies its total universality property.

Background

The paper distinguishes the universality property, which applies to generic pseudo-generic reals, from the total universality property, which applies to all pseudo-generic reals. The authors prove that these notions coincide under suitable correctness assumptions, but leave open whether the implication holds for Borel relations without those assumptions.

References

Does the universality property of a Borel relation $R$ already imply the total universality property?

— Universal domination and idealized forcing  (2608.18964 - Schilhan, 19 Aug 2026) in Question immediately following Lemma in Section 3, subsection “On genericity”

Is true eventually different forcing proper somewhere? Is there some condition $B$, such that each eventually different real $d \in B$ over a model $M \ni B$ is universally eventually different over $M$?

— Universal domination and idealized forcing  (2608.18964 - Schilhan, 19 Aug 2026) in Question in Section 7, subsection “True eventually different forcing”