The palindromic surjective-strategy conjecture for wreath products

Let $G\wr H$ be a wreath product, and call a surjective strategy palindromic when its sequence of moves $S'=\{k'_i\in K\}_{i=1}^N$ satisfies $k'_i=k'_{N-i+1}$ for every $i$. Palindromic strategy conjecture. Whenever $G\wr H$ has a surjective strategy, it also has a palindromic surjective strategy. The conjecture is motivated by the fact that all examples known to the authors admit such a strategy; the source notes that a counterexample might occur for a trivial wreath product.