Supporting-tree decomposition conjecture for dyadic rearrangements

Let $\t$ be a rearrangement of the dyadic intervals, and write $\cF_1,\dots,\cF_N$ for subcollections of the dyadic intervals. Suppose there is a constant $C$ such that, for every dyadic interval $I$,

$$\left|\bigcup_{J\sbe I}\t(J)\right|\leq C|I|\quad\text{and}\quad\left|\bigcup_{J\sbe I}\t^{-1}(J)\right|\leq C|I|.$$

A collection admits a supporting tree if there are measurable sets $A_I$ forming a tree, with uniformly controlled measure and uniformly positive intersections with both $I$ and $\t(I)$. Supporting-tree decomposition conjecture. Do these two inequalities imply that the entire collection of dyadic intervals can be decomposed into $\cF_1,\dots,\cF_N$, with $N=N(C)$, so that each restriction $\t:\cF_i\to\t(\cF_i)$ admits a supporting tree? Such a decomposition would provide a combinatorial route to vector-valued boundedness for rearrangement operators; the supplied text does not state whether this problem has been resolved.