The convex-radii cover conjecture for metric trees

Let $(r_1,\dots,r_k)$ be a sequence of non-negative real numbers satisfying

$$r_1\leq r_2-r_1\leq r_3-r_2\leq\cdots\leq r_k-r_{k-1},$$

so that $(r_1,\dots,r_k)$ is convex. Let $T$ be a metric tree with total length $|T|\leq 2\sum_{i=1}^k r_i$. Convex-radii cover conjecture. The sequence $(r_1,\dots,r_k)$ is a cover of $T$. This conjecture is a metric analogue of the Burning Number Conjecture and includes the sequence $(1,\ldots,k)$. The source presents it as a broad natural class for which the covering property should hold; its general status is not resolved there.