The tree domination-game pair conjecture

Let $T$ be a tree, and let $\gamma_g(T)$ and $\gamma_g'(T)$ denote the Dominator-start and Staller-start game domination numbers, respectively. A pair $(r,s)$ is realizable by a tree if there is a tree $T$ with $(\gamma_g(T),\gamma_g'(T))=(r,s)$.

Tree domination-game pair conjecture. No pair of the form $(k,k-1)$ can be realized by a tree.

The paper proves that $(3,2)$ and $(4,3)$ are not realizable and establishes realizability for the other discussed families, while the general nonrealizability of $(k,k-1)$ remains conjectural.