Are there non-trivial equivalent characterizations of $\delta$-strongly compact (and almost strongly compact) cardinals in terms of generalized tree properties?

Recall the definitions as per Joan Bagaria and Menachem Magidor, *On $\omega_1$-strongly compact cardinals*, JSL 79 (2014), 266-278.

Suppose $\delta > \omega$ is a cardinal. A cardinal $\kappa > \delta$ is *$\delta$-strongly compact* if for every set $I$, every $\kappa$-complete filter on $I$ can be extended to a $\delta$-complete ultrafilter on $I$. A limit cardinal $\kappa > \omega$ is *almost strongly compact* if $\kappa$ is $\delta$-strongly compact for every uncountable cardinal $\delta < \kappa$. Note that if $\lambda > \kappa$ and $\kappa$ is $\delta$-strongly compact, then $\lambda$ is also $\delta$-strongly compact, although, apparently serving the interests of Bavel, in the literature there are definitions going in the opposite direction, as noted in Joel's comment.

My question is whether there are localized variants of strong and super tree properties that are equivalent to the parametrized large cardinal concepts. For example, can $\aleph_2$-strong compactness be characterized in terms of a strong tree property hovering around $\aleph_2$? The general project is the search for large cardinal properties that small cardinals may possess, once inaccessibility is dropped.

Notice that by a theorem of Magidor, it is consistent that the first $\omega_1$-strongly compact is also the first measurable, in which case it is strongly compact, so will satisfy the strong tree property. But there is also a model of ZFC in which the first $\omega_1$-strongly compact is singular.

The definitions of those generalized tree properties are given at the end of the question, below.

The context is the results of Di Prisco, Erdös, Jech, Magidor, Tarski, Weiss and Zwicker, who variously prove for the parameter-free concepts:

**Theorem**

Suppose $\kappa$ is inaccessible.

TFAE: (i) $\kappa$ is weakly compact; (ii) $\kappa$ has the tree property; (iii) $\kappa \rightarrow (\kappa)^{2}_{2}$.

TFAE: (i) $\kappa$ is strongly compact; (ii) $\kappa$ has the strong tree property.

TFAE: (i) $\kappa$ is supercompact; (ii) $\kappa$ has the super tree property.

Now for the definitions of the relevant tree properties, taken from Laura Fontanella, *The strong tree property at successors of singular cardinals*, JSL 79 (2014), 193-207.

Suppose $\kappa \geq \aleph_2, \lambda \geq \kappa, \lambda \in Ord$. A *$(\kappa, \lambda)$-tree* is a set $F$ with the properties: (1) for every $f \in F, f:X \rightarrow 2$, for some $X \in [\lambda]^{<\kappa}$; (2) for all $f \in F$, if $X \subseteq dom(f)$, then $f \vert X \in F$; (3) $Lev_X(F) = \lbrace f \in F : dom(f) = X \rbrace \neq \emptyset$ for all $X \in [\lambda]^{<\kappa}$; (4) $\vert Lev_X(F) \vert < \kappa$ for all $X \in [\lambda]^{<\kappa}$.

Note $(\kappa, \lambda)$-trees are not actually trees, despite the terminology: for $f \in Lev_X(F), \lbrace f \vert Y : Y \subseteq X \rbrace$ is not well-ordered in general.

A *cofinal branch* is a function $b:\lambda \rightarrow 2$ such that $b \vert X \in Lev_X(F)$ for all $X \in [\lambda]^{<\kappa}$.

An *$F$-level sequence* is a function $D: [\lambda]^{<\kappa} \rightarrow F$ such that $D(X) \in Lev_X(F)$ for every $X \in [\lambda]^{<\kappa}$.

An *ineffable branch* for an $F$-level sequence $D$ is a cofinal branch $b$ such that $\lbrace X \in [\lambda]^{<\kappa} : b \vert X = D(X) \rbrace$ is stationary.

The cardinal $\kappa$ has the *strong tree property* if every $(\kappa, \lambda)$-tree has a cofinal branch for every ordinal $\lambda \geq \kappa$.

A cardinal $\kappa$ has the *super tree property* if for every ordinal $\lambda \geq \kappa$, every $F$-level sequence $D$ in every $(\kappa, \lambda)$-tree has an ineffable branch.

The case of $\aleph_2$ is of interest, much as the classical $\aleph_2$-tree property has been a catalyst of much research and the invention of new forcing (Mitchell); similarly, tree properties at successors of singular cardinals present challenges taken up by many mathematical logicians (Neeman, Unger, Fontanella, Weiss, Cummings, Foreman, Sinapova, Magidor, Shelah, and others).

It is conjectured (by Fontanella) that the successor of a singular limit of supercompact cardinals has the super tree property. In the paper quoted above, she proved that if $\kappa$ is a singular limit of strongly compact cardinals, then $\kappa^+$ has the strong tree property.