The probability-measure cover conjecture for metric trees

Let $r>0$, let $T$ be a metric tree with total length $|T|\geq 2r$, and let $\mathcal{C}(T,0)$ denote the relevant space of covers of $T$ by radius-zero centers. Let $E\nu$ denote the expected cover associated with a probability measure $\nu$. Probability-measure cover conjecture. There exists a probability measure $\nu$ on $\mathcal{C}(T,0)$ such that

$$E\nu\leq \frac{|T|}{r}\boldsymbol{U}[0,r].$$

The conjecture is proposed as the only obstruction to eliminating the error term in the paper's asymptotic result. It is proved in the source for metric trees with at most three leaves, but remains open in general.