Infinite-dimensionality conjecture for unimodular trees with infinitely many ends

Let $[\boldsymbol{T},\boldsymbol{o}]$ be a unimodular tree with infinitely many ends, endowed with its graph-distance metric. Infinite-dimensionality conjecture. One has

$$\operatorname{udim}_H(\boldsymbol{T})=\infty.$$

This is identified as the main conjecture of the subsection and is stated to follow from the conjecture for the 3-regular tree; special cases are discussed and proved later.