Almost Kurepa Suslin trees and destructibility of the Guessing Model Property

Abstract: Building on recent work of Krueger and the second author, we prove the consistency of the Guessing Model Principle at $ω_2$ together with the existence of an almost Kurepa Suslin tree. In particular, it is consistent that the Guessing Model Principle holds but is destructible by a ccc forcing of size $ω_1$. We also prove the consistency of the existence of a weak Kurepa tree together with the failure of the Kurepa Hypothesis and a certain guessing model principle that, for example, implies the tree property at $ω_2$.