Very weak square and good points

Question

This is probably well known but I'll appreciate pointers to references: Is there any model where for a singular cardinal $\kappa$ of cofinality $\omega$, Very Weak Square holds at $\kappa$ but every scale on $\kappa$ has stationarily many bad points of cofinality $\omega_1$?

Definitions: Very Weak Square at $\kappa$: there is a sequence $\langle C_\alpha: \alpha<\kappa^+\rangle$ where for a club of $\alpha\in \kappa^+$

1. $C_\alpha\subset \alpha$ is unbounded
2. for any bounded $x\in [C_\alpha]^\omega$, there is $\beta<\alpha$ such that $x=C_\beta$.

Scales at $\kappa:$ For some $\langle \kappa_i: i\in \omega\rangle$ increasing cofinal regular cardinals, $\langle f_\alpha\in \prod_{i\in \omega} \kappa_i: \alpha<\kappa^+\rangle$ is a scale if it is increasing and cofinal in $<^*$ (increasing mod finite). $\alpha<\kappa^+$ is a good point if $cf(\alpha)>\omega$ and there is $A\subset \alpha$ cofinal and $m\in \omega$ such that $\{f_\alpha\restriction_{>m}: \alpha\in A\rangle$ is pointwise increasing. $\alpha$ is bad if it's not good.

It is known that the assertion is false at $\aleph_\omega$ (Foreman-Magidor) as VWS at $\aleph_\omega$ implies there is a club $C\subset \aleph_{\omega+1}$ such that all $\alpha\in C\cap cof(\omega_1)$ is good. But the proof goes through an intermediate principle (approachability) and VWS does not in general imply AP at larger singular cardinals.

Answer 1

Though Very Weak Square at $\kappa$ does not in general imply $\mathrm{AP}_\kappa$, it does imply that $\kappa^+ \cap \mathrm{cof}(\omega_1)$ is in the approachability ideal on $\kappa^+$. In fact, the Very Weak Square sequence $\langle C_\alpha \mid \alpha < \kappa^+ \rangle$ is itself a witness to this, provided that we have $\mathrm{otp}(C_\alpha) = \omega_1$ for all $\alpha \in \kappa^+ \cap \mathrm{cof}(\omega_1)$, which can always be arranged. It then follows from the usual arguments, as in the Foreman-Magidor paper on Very Weak Square, that, in a scale on $\kappa$, the set of bad points of cofinality $\omega_1$ is non-stationary.