Composition-dependence conjecture for maps between resonance complexes

For every $n\geq 1$, let $\operatorname{Res}\,\lambda$ denote the resonance set associated with a number partition $\lambda$ of length $n$. If $\operatorname{Res}\,\tilde\lambda\geq\operatorname{Res}\,\lambda$, choose compositions $\tilde\alpha$ and $\alpha$ of types $\tilde\lambda$ and $\lambda$, respectively, whose resonance sets satisfy the required inclusion. This choice induces a map

$$\gamma(\tilde\alpha,\alpha):\delta_{\tilde\lambda}\rightarrow\delta_\lambda$$

and hence a homology map

$$\gamma(\tilde\alpha,\alpha)_*:\widetilde H_*(\delta_{\tilde\lambda})\rightarrow\widetilde H_*(\delta_\lambda).$$

Composition-dependence conjecture. The isomorphism type of $\gamma(\tilde\alpha,\alpha)_*$ depends not only on the actual number partitions $\tilde\lambda$ and $\lambda$, rather than their sets of resonances, but even on the choice of the pair of compositions $\alpha$ and $\tilde\alpha$. Thus the induced homology map is not determined solely by the resonance data or by the number partitions.