ULCFS and strong ULCFS conjectures for NTP$_2$ and resilient formulas

Let $\varphi(x,y)$ be a partitioned formula. A formula is NTP$_2$ when it does not have the tree property of the second kind, and it is resilient in the sense used in the paper. ULCFS and strong ULCFS conjecture. If $\varphi(x,y)$ is NTP$_2$, or if the theory $T$ is NTP$_2$, then $\varphi(x,y)$ satisfies ULCFS. If $\varphi(x,y)$ is resilient, or if $T$ is resilient, then $\varphi(x,y)$ satisfies strong ULCFS. The paper establishes related implications, including that strong ULCFS implies resilience and that ULCFS implies NTP$_2$; the converses proposed here remain open.