Equivalence characterization for the tree σE

Let $E$ be a linearly ordered set, and let $\sigma E$ be the tree of all bounded well-ordered subsets of $E$, ordered by initial segment inclusion. The following conditions are equivalent:

Equivalence conjecture for $\sigma E$.

$$\begin{aligned} &\text{(1) Neither }\omega_1\text{ nor }\mathbb Q\text{ as linearly ordered sets are contained in }E;\\ &\text{(2) }\sigma E\text{ is a special tree};\\ &\text{(3) }\sigma E\text{ is a }Q\text{-space};\\ &\text{(4) }\sigma E\text{ is a }\Delta\text{-space}. \end{aligned}$$

This proposed equivalence is presented as a provisional result conditional on the conjecture that $\sigma\mathbb Q$ is not a $\Delta$-space. The source does not establish that conjecture, and therefore the equivalence remains open.