Concentration conjecture for the path-covering number in random trees

Let $T$ be the random tree under consideration, let $t$ denote its size parameter, and let $\operatorname{cov}_\ell(T)$ be the minimum number of paths of length at most $\ell$ needed to cover $T$. Suppose that $\ell=\ell(t)$ satisfies

$$\ell=o(\sqrt{t})$$

as $t$ tends to infinity. Concentration conjecture. For every $\delta>0$,

$$\left|\operatorname{cov}_\ell(T)-\mathbb{E}[\operatorname{cov}_\ell(T)]\right|<\frac{\delta t}{\ell}$$

with high probability. The preceding argument establishes concentration in a more restricted range, while the conjecture proposes that the weaker condition $\ell=o(\sqrt{t})$ suffices; it remains open in the supplied source.