Uniqueness conjecture for power domination reconfiguration graphs of complete bipartite graphs

Let $K_{s,t}$ be the complete bipartite graph with part sizes $s$ and $t$, and let $\mathscr{PD}^{TAR}(K_{s,t})$ denote its power domination reconfiguration graph under token addition/removal reconfiguration. Uniqueness conjecture. $\mathscr{PD}^{TAR}(K_{s,t})$ is unique if and only if $s\ne 2$ and $t\ne 2$. This is supported by the uniqueness results for $K_{1,t}$ with $t\ge 3$ and $K_{3,3}$, together with Sage computations for $s\in\{3,4\}$ and $t\in\{3,4,5\}$; the conjecture proposes the complete characterization, including the cases not covered by those results.