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This question follows up on Bound on queries to a tree with unusual probabilities, where @fedja was able to disprove my conjecture under only constraints (1-4) below. I restate the relevant facts here ...

Consider a tree $\mathcal{T}(r) = (V,E)$ rooted at $r \in V$. Let $\kappa_r: V \longrightarrow [0,1]$ such that $\sum_{v \in V} \kappa_r(v)^2 = 1$. Furthermore, for any given vertex $v \neq r$, $\...