The minimal surjective-strategy conjecture for wreath products

Let $G\wr H$ be a wreath product with state group $K$, and call a surjective strategy minimal when it has the form $\{k_i\in K\}_{i=1}^{|K|-1}$. Minimal strategy conjecture. Whenever $G\wr H$ has a surjective strategy, it also has a minimal surjective strategy $\{k_i\in K\}_{i=1}^{|K|-1}$. Minimal strategies would give the shortest possible guaranteed bound discussed in the source; the conjecture is based on the examples known to the authors and remains open.