Remove polylogarithmic slack in attainable two-point rates

Let $\mathcal{F}$ be the class of all probability distributions $P$ on $\mathbb{R}$ such that $P$ has a density of the form

where $\mu\in\mathbb{R}$, $\Pi$ is a probability measure on an index set $\Theta$, and for every $\theta\in\Theta$, $g_\theta$ is a probability density on $\mathbb{R}$ that is symmetric and log-concave about $0$ (equivalently, $g_\theta(t)=g_\theta(-t)$ for all $t$, and $\log g_\theta$ is concave on its support). Then each component has mean $0$, so $\mu(P):=\mathbb{E}_P[X]=\mu$ is well-defined. In the adaptive estimation model, one observes $X_1,\dots,X_n \stackrel{\mathrm{i.i.d.}}{\sim} P$ with unknown $P\in\mathcal{F}$, and an estimator is any measurable map $\hat\mu_n:\mathbb{R}^n\to\mathbb{R}$.

For distributions $P,Q$ on $\mathbb{R}$, define squared Hellinger distance by $H^2(P,Q):=\frac12\int(\sqrt{dP/d\nu}-\sqrt{dQ/d\nu})^2\,d\nu$, where $\nu$ dominates both $P$ and $Q$. For $m\in\mathbb{N}$, define the local Hellinger modulus at $P$ by

(The constant $1/3$ can be replaced by any fixed number in $(0,1)$ at the cost of universal constant changes.)