Consistency of PSR for general stationary sets $S\subset [H_\kappa]^\omega$

Question

Feng-Jech's Projective-Stationary Reflection (PSR): For every regular $\kappa\geq\omega_2$, if $S\subset[H_\kappa]^\omega$ is projective-stationary, then there is an increasing continuous $\in$-chain $\langle N_\alpha\mid \alpha<\omega_1\rangle$ of countable elementary submodels of $H_\kappa$ such that $N_\alpha\in S$ for all $\alpha<\omega_1$.

PSR follows from MM: Given stationary $S\subset[H_\kappa]^\omega$, there is a natural forcing $P_S$ to shoot an increasing continuous $\in$-chain of length $\omega_1$, and $S$ is projective-stationary iff $P_S$ is SSP. Not every stationary set is projective-stationary.

Now replace, in the statement of PSR, the word "projective-stationary" by "stationary", namely:
For every regular $\kappa\geq\omega_2$, if $S\subset[H_\kappa]^\omega$ is stationary, then there is an increasing continuous $\in$-chain $\langle N_\alpha\mid \alpha<\omega_1\rangle$ of countable elementary submodels of $H_\kappa$ such that $N_\alpha\in S$ for all $\alpha<\omega_1$.
Is this principle known to be consistent or inconsistent?

Answer 1

This principle is inconsistent: Let $T\subseteq\omega_1$ be stationary such that $T$ does not contain a club (such a set exists e.g. by Solovay splitting which is certainly overkill). Let for an arbitrary $\kappa\geq\omega_2$ the set $S\subseteq[H_{\kappa}]^{\omega}$ consist of those countable $M\prec H_{\kappa}$ such that $\omega_1\in M$ and $M\cap\omega_1\in T$. $S$ is stationary in $[H_{\kappa}]^{\omega}$ since for any club $C\subseteq[H_{\kappa}]^{\omega}$ the set $C':=\{M\cap\omega_1\;|\;M\in C\}$ contains a club in $[\omega_1]^{\omega}=\omega_1$ (this is due to Menas, see "On Strong Compactness and Supercompactness").

However, there is no chain as you require: If $(N_{\alpha})_{\alpha<\omega_1}$ is an increasing and continuous chain of elements of $S$ then the sequence $(N_{\alpha}\cap\omega_1)_{\alpha<\omega_1}$ is increasing (since $N_{\alpha},\omega_1\in N_{\alpha+1}$ implies $N_{\alpha}\cap\omega_1\in N_{\alpha+1}$ and thus $N_{\alpha}\cap\omega_1\in N_{\alpha+1}\cap\omega_1$) and continuous (clearly). However, then $\{N_{\alpha}\cap\omega_1\;|\;\alpha<\omega_1\}$ has to be club in $\omega_1$ (it has size $\omega_1$, so it is unbounded) and is contained in $T$, a contradiction.