Sublinear polynomial upper bound for the on-line ranking number of trees

Let $T$ be an $n$-vertex tree with maximum degree $k$, and let $\mathring{\rho}(T)$ denote its on-line ranking number.

On-line ranking conjecture. There exist universal constants $a$ and $b$ satisfying $0<a<1<b$ such that

$$\mathring{\rho}(T)\leq b(kn)^a.$$

This conjecture proposes a general upper bound for the on-line ranking number of trees, analogous to the bounds established for the trees $T_{k,d}$ in the paper. The supplied context does not state whether the conjecture has been resolved.