Strong $m$-ary sensitivity conjecture

Let $m$ be a positive integer, let $\varepsilon$ be an $m$-th primitive root of unity, and let $H(n,m)$ be the Hamming graph. Write $H_1,\dots,H_m$ for possibly empty induced subgraphs whose vertex sets partition the vertices of $H(n,m)$, and let $\Delta(H_i)$ denote the maximum degree of $H_i$. Strong $m$-ary sensitivity conjecture. There exists $\mu>0$ such that, whenever

$$\sum_{i=1}^{m}|V(H_i)|\varepsilon^i\neq 0,$$

one has

$$\max\{\Delta(H_1),\dots,\Delta(H_m)\}\in\Omega(n^\mu).$$

The paper presents this as a stronger, more natural open reformulation of the $m$-ary sensitivity conjecture. Its claimed strength is supported by the paper's discussion, while the relationship to the preceding formulation is the subject of the stated context.