The nonassociative loop-switch strategy conjecture

Let $\mathcal{L}=(\{1,a,b,c,d\},\ast)$ be the smallest loop that is not a group, with multiplication operation $\ast$. Loop-switch strategy conjecture. There exists a nontrivial adversarial group $H$ such that the generalized spinning switches puzzle with switches behaving like $\mathcal{L}$ has a winning strategy for the puzzle-solver. This asks whether the group-based framework extends to nonassociative switches; the source introduces it as an open question and gives no candidate group $H$.