The orbit-bound conjecture for shortest surjective strategies

Let $K/H$ be the set of equivalence classes of $K$ under the action of $H$, and let $N$ denote the length of a surjective strategy. Orbit-bound conjecture. If $G\wr H$ has a surjective strategy, then it has a surjective strategy of length $N<2^{|K/H|-1}$. This is a weaker alternative to the minimal-strategy conjecture, providing an exponential upper bound in the number of $H$-orbits; the source leaves it open.