Noncanonicity conjecture for homology maps between resonance complexes

Let $\lambda$, $\mu$, and $\tilde\mu$ be number partitions. For number partitions $\lambda>\mu$, the inclusion $i(\mu,\lambda):\delta_\mu\rightarrow\delta_\lambda$ induces a homomorphism

$$i(\mu,\lambda)_*:\widetilde H_*(\delta_{\,\operatorname{Res}\,\mu})\rightarrow\widetilde H_*(\delta_{\,\operatorname{Res}\,\lambda}).$$

Noncanonicity conjecture. There exist number partitions $\lambda$, $\mu$, and $\tilde\mu$ such that $\lambda>\mu$, $\lambda>\tilde\mu$, $\operatorname{Res}\,\mu=\operatorname{Res}\,\tilde\mu$, and the homomorphisms $i(\mu,\lambda)_*$ and $i(\tilde\mu,\lambda)_*$ induced by the respective inclusion maps are nonisomorphic. The conjecture asserts that the induced map depends on more than the resonance sets of its source and target, so no unique map $i(\operatorname{Res}\,\mu,\operatorname{Res}\,\lambda)_*$ can be defined.