Homogeneity of dominating forcing

Determine whether the forcing $\mathbb{P}_{<^*}$ is homogeneous or inhomogeneous, and characterize its global forcing structure beyond the condition of strictly increasing functions.

Background

The paper proves that dominating forcing induced by eventual dominance is proper and has the total universality property. Earlier work established an equivalence with Laver forcing below the condition of strictly increasing functions, but the authors do not obtain such an equivalence globally and explicitly leave the homogeneity question unresolved.

References

We do not see an argument that this must hold everywhere, nor do we know how homogeneous or inhomogeneous $\mathbb{P}_{<*}$ is.

— Universal domination and idealized forcing  (2608.18964 - Schilhan, 19 Aug 2026) in Paragraph following Corollary in Section 6, subsection “Dominating forcing”