It is a theorem of Baumgartner and Laver that iterating Sacks forcings of weakly compact length gives rise to the tree property at $\omega_2$. Natural questions (at least for me) are: do we get stronger tree properties at $\omega_2$ if we start with larger cardinals? In particular, if the length is strongly compact, do we get the strong tree property at $\omega_2$? If the length is supercompact, do we get the super tree property at $\omega_2$? I suspect it is known so I appreciate pointers to references.
For uncountable cardinals $\kappa\leq \lambda$ with $\kappa$ being regular, we say $F\subset \bigcup_{u\in [\lambda]^{<\kappa}} 2^u$ is a $(\kappa,\lambda)$-tree if it is closed under taking restrictions, and for each $u\in [\lambda]^{<\kappa}$, $|F_u:=\{f\in F: dom(f)=u\}|<\kappa$. A function $d: \lambda\to 2$ is a cofinal branch if for all $u\in [\lambda]^{<\kappa}$, $d\restriction u\in F_u$. Given $\bar{f} = \langle f_u\in F: dom(f_u)=u\rangle$ (called a level sequence), we say $d: \lambda\to 2$ is an $\bar{f}$-ineffable branch if $\{u\in [\lambda]^{<\kappa}: d\restriction u = f_u\}$ is stationary in $[\lambda]^{<\kappa}$.
We say $\kappa$ has the strong tree property if for any $\lambda\geq \kappa$, any $(\kappa, \lambda)$-tree has a cofinal branch.
We say $\kappa$ has the super tree property if for any $\lambda\geq \kappa$, any $(\kappa, \lambda)$-tree $F$ and any level sequence $\bar{f}$, there exists an $\bar{f}$-ineffable branch.
Edit: I just realized the answer in the case where the length is strongly compact is yes: the countable support iteration of Sacks forcings of strongly compact length forces the Semistationary Reflection Principle (equivalent to the statement that all stationary preserving forcings are semiproper and in fact stronger statement, i.e. the Baire version of Rado's conjecture, holds), along with $\neg CH$, implies the strong tree property holds at $\omega_2$ by Torres-Perez and Wu (https://www.researchgate.net/publication/306394279_Strong_Chang's_Conjecture_Semi-Stationary_Reflection_the_Strong_Tree_Property_and_two-cardinal_square_principles). The question remains for the case where the length is supercompact.
