Eternal distance-$k$ domination versus ordinary domination in trees

Let $T$ be a tree and let $k>2$. Write $\gamma_k(T)$ for the distance-$k$ domination number, $\gamma_{all,k}^{\infty}(T)$ for the eternal distance-$k$ domination number, and $\gamma_{\left\lfloor k/2\right\rfloor}(T)$ for the domination number at distance $\left\lfloor k/2\right\rfloor$. Eternal distance-$k$ domination conjecture. If

$$\gamma_k(T)=\gamma_{all,k}^{\infty}(T),$$

then

$$\gamma_{all,k}^{\infty}(T)=\gamma_{\left\lfloor k/2\right\rfloor}(T).$$

This generalizes the analogous implication established in the paper for $k=2$ and is presented as an open problem for $k>2$; the general case remains unresolved.