Is Interaction Necessary for Order-Optimal 1-bit Mean Estimation?

Fix $k>1$. For known parameters $\lambda\geq\sigma>0$, consider the nonparametric family

$$\mathcal{D}(k,\lambda,\sigma) :=\left\{D: \mu(D):=\mathbb{E}_{X\sim D}[X]\in[-\lambda,\lambda],\quad \mathbb{E}_{X\sim D}\!\left[|X-\mu(D)|^k\right]\leq\sigma^k \right\}.$$

Apart from this moment condition, a distribution $D$ may be discrete, asymmetric, or supported on an unbounded set. The parameter $\lambda$ is a search radius for the mean, while $\sigma$ represents the intrinsic scale.

A 1-bit protocol observes independent samples $X_1,\ldots,X_n\sim D$ only through messages

$$Y_t=\mathbf{1}\{X_t\in A_t\},$$

where each $A_t\subseteq\mathbb{R}$ is measurable. In an adaptive protocol, $A_t$ may be chosen from the past transcript $(A_1,Y_1,\ldots,A_{t-1},Y_{t-1})$ and internal randomness. In a fully non-adaptive protocol, all sets $A_1,\ldots,A_n$ are fixed before any messages are observed, possibly using public or private randomness. For $\epsilon>0$ and $\delta\in(0,1/2)$, a protocol is $(\epsilon,\delta)$-accurate over $\mathcal{D}(k,\lambda,\sigma)$ if its output $\widehat\mu$ satisfies

$$\sup_{D\in\mathcal{D}(k,\lambda,\sigma)} \Pr\!\left\{|\widehat\mu-\mu(D)|>\epsilon\right\}\leq\delta,$$

where the probability is over the samples and any internal randomness.

For $\epsilon$ below a sufficiently small constant multiple of $\sigma$, define the adaptive 1-bit minimax rate, up to constants depending only on $k$, by

$$r_k(\lambda,\sigma,\epsilon,\delta) :=\log\frac{\lambda}{\sigma}+ \begin{cases} \dfrac{\sigma^2}{\epsilon^2}\log\dfrac{1}{\delta}, & k>2,\\[0.8em] \dfrac{\sigma^2}{\epsilon^2}\log\dfrac{\sigma}{\epsilon}\log\dfrac{1}{\delta}, & k=2,\\[0.8em] \left(\dfrac{\sigma}{\epsilon}\right)^{\!k/(k-1)}\log\dfrac{1}{\delta}, & 1<k<2. \end{cases}$$

Do there exist constants $c_k,C_k>0$ such that, for all $\lambda\geq\sigma>0$, all $0<\epsilon\leq c_k\sigma$, and all $\delta\in(0,1/2)$, there is a fully non-adaptive 1-bit protocol that is $(\epsilon,\delta)$-accurate over $\mathcal{D}(k,\lambda,\sigma)$ using at most

$$n\leq C_k r_k(\lambda,\sigma,\epsilon,\delta)$$

samples?