Polynomial-time complexity of distance-$d$ independent set reconfiguration on trees under token sliding

Let $d\geq 3$. In distance-$d$ independent set reconfiguration, denoted by $\textsc{D}d\textsc{ISR}$, configurations are distance-$d$ independent sets, and under the token-sliding rule, denoted by $\sfTS$, one token may slide along an edge in each move. The input graph is a tree. Polynomial-time reconfiguration conjecture. $\textsc{D}d\textsc{ISR}$ under $\sfTS$ on trees is in $\ttP$ for $d\geq 3$. The complexity of this problem is stated in the source to remain unknown, so this is an open problem concerning reconfiguration on trees.