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For any $$m\ge \mathrm{3\$}$ we show that the Hamming graph $$H(n,m)\$$ admits an imbalanced partition into $m$ sets, each inducing a subgraph of low maximum degree. This improves previous results by Tandya and by Potechin and Tsang, and disproves the Strong $$\mathrm{m\$}$-ary Sensitivity Conjecture of Asensio, García-Marco, and Knauer. On the other hand, we prove their weaker $$\mathrm{m\$}$-ary Sensitivity Conjecture by showing that the sensitivity of any $$\mathrm{m\$}$-ary function is bounded from below by a polynomial expression in its degree.