Relative consistency of quagmires without morasses

Determine whether it is consistent that a (ω_1, Lim(ω_1))-quagmire exists while no (ω_1, 1)-morass exists.

Background

The paper constructs a (κ, Lim(κ))-quagmire from a simplified (κ, 1)-morass and observes that it is unknown whether, at ω_1, this quagmire is genuinely weaker than a morass. A prior result gives a model with an ω_1-quagmire, meaning a (ω_1, ∅)-quagmire, but no (ω_1, 1)-morass; this does not settle the stronger Lim(ω_1) version. The recorded question asks for the relative consistency of the stronger quagmire together with the failure of an ω_1-morass.

References

Finally, we do not know whether the existence of a $(\omega_1, \mathrm{Lim}(\omega_1))$-quagmire is genuinely weaker than the existence of a $(\omega_1, 1)$-morass. Note that Komj{a}th constructed, assuming the existence of a Mahlo cardinal, a model of $\mathsf{ZFC}$ in which there is an $\omega_1$-quagmire (i.e., a $(\omega_1, \emptyset)$-quagmire in our notation) but no $(\omega_1, 1)$-morass. \begin{question} Is it consistent that there is a $(\omega_1, \mathrm{Lim}(\omega_1))$-quagmire but no $(\omega_1, 1)$-morass? \end{question}

— Quagmires and large Suslin forests  (2608.28505 - Notaro, 28 Aug 2026) in Section 4, “Questions”