Squares, ascent paths, and chain conditions, part 2

We continue our pair of talks on connections between square principles, trees with ascent paths, and strong chain conditions.
In the previous talk, we discussed trees with ascent paths; we turn our attention this week to chain conditions. In particular, we will prove that, if $\kappa > \aleph_1$ is a regular cardinal and $\square(\kappa)$ holds, then:

1) There is a $\kappa$-Knaster poset $\mathbb{P}$ such that $\mathbb{P}^\omega$ is not $\kappa$-c.c.

2) There is a $\kappa$-Knaster poset $\mathbb{P}$ that is not $\kappa$-stationarily layered.

This talk will rely only minimally on material from the previous talk, and the results are joint work with Philipp Lücke.