Spectral-measure decay conjecture for uniformly discrete invariant random subgroups

Let $G$ be the Lie group under consideration, let $k$ be a degree, and let an IRS $\mu$ of $G$ be $\varepsilon$-discrete in the sense used in the source. Let $\nu_\mu^k$ be the spectral measure associated with the degree-$k$ Laplace operator. Spectral-measure decay conjecture. For every $k$ and every $\varepsilon>0$, there exist positive constants $C$ and $\eta_0$ such that every $\varepsilon$-discrete IRS $\mu$ satisfies

$$\nu_\mu^k((0,\eta])\leq\frac{C}{\log(1/\eta)}\quad(0<\eta<\eta_0).$$

Such a bound would prevent spectral mass from accumulating too rapidly near zero and would yield pointwise convergence at zero in situations where weak convergence alone is insufficient.