The asymptotically better-than-random strategy conjecture

Let $K$ be the state group of a generalized spinning switches puzzle, and measure performance by the expected number of moves when the initial hidden state is chosen uniformly. Asymptotic improvement conjecture. There exists a constant $\frac{1}{2}<c<1$ such that, for all finite wreath products $G\wr H$ with sufficiently large $|K|$, the expected number of moves is less than $c|K|$. The conjecture formalizes the claim that the puzzle-solver can always do asymptotically better than randomly guessing; the source gives no proof or resolution.