Water Transport on 3-Spiders: Unbounded Lengths of Shortest Optimal Strategies

Let $G$ be a finite simple graph. An averaging operation selects a nonempty connected vertex set and replaces the weights of its vertices by their arithmetic mean. For a nonnegative initial weight function $ω:V(G)\to\mathbb R_{\ge0}$ and a target vertex $x$, let $t(G,x,ω)$ denote the minimum number