The anti-concentration norm-sum conjecture for inhomogeneous random walks

Let $G$ be the group and let $\mu_i$ be the measures appearing in Theorem 3.1, with $A_i$ denoting their supports and $id_G\in A_i$. Assume that

$$\rho(\mu_1,\dots,\mu_n)\ge n^{-O(1)}.$$

Anti-concentration norm-sum conjecture. There exist consecutive indices $i_0,\dots,i_0+n^{1-\varepsilon}$ such that

$$\sum_{i_0\leq i\leq i_0+n^{1-\varepsilon}}\sum_{a_i\in A_i}\|a_i\|_{HP,X}^2\ll 1.$$

The conjecture proposes that the waste from using the triangle inequality can be avoided over a block of consecutive steps, yielding a uniformly bounded total squared $HP$-norm. The supplied text gives no resolution status or further evidence, so the conjecture is recorded as open.