Let $p \in (0,1)$. Take $E$ to be the edge set of the trivalent tree $T$, and $G$ to be the automorphism group of $T$. Let $f$ be any $G$-equivariant map from the measure space $([0,1]^E, \text{d}x^{\otimes E})$ to $(\{0,1\}^E, \text{Bernoulli}(p)^{\otimes E})$. Is it necessarily the case that there is some measurable space $A$, some $G$-invariant probability measure $m$ on $A^E$ and some $G$-equivariant isomorphism

$$h : ([0,1]^E, \text{d}x^{\otimes E}) \to (\{0,1\}^E \times A^E, \text{Bernoulli}(p)^{\otimes E} \times m)$$

such that $f = \text{proj}_1 \circ h$ ?

$G$ acts by shift on $X^E$ and diagonally on $\{0,1\}^E \times A^E$.