Ordinary posterior contraction for heavy-tailed neural-network priors

Observe independent pairs $(X_i, Y_i)$, $i = 1, \dots, n$, with $$X_i \sim P_X, \qquad Y_i = f_0(X_i) + \varepsilon_i, \qquad \varepsilon_i \sim N(0,1),$$ where $P_X$ is a probability distribution on $[0,1]^d$ and the errors are independent of the design; the noise variance is known. Fix $\delta > 0$. Let $\Pi_n$ be the law of a fully connected ReLU network on $[0,1]^d$ with $L_n = \lceil (\log n)^{1+\delta} \rceil$ hidden layers, each of width $\lceil \sqrt{n} \rceil$, and an affine scalar output, whose weights and biases are independent copies of $\sigma_n Z$ with $\sigma_n = \exp\{-(\log n)^{2(1+\delta)}\}$. Here $Z$ has a density $h$ that is positive, bounded, symmetric and nonincreasing on $[0, \infty)$, and for some $c_1, c_2 > 0$ and $\kappa \ge 0$, $$\log\frac{1}{h(x)} \le c_1\bigl[1 + \{\log(1+x)\}^{1+\kappa}\bigr] \quad (x \ge 0), \qquad \int_x^\infty h(u)\,du \le \frac{c_2}{x} \quad (x \ge 1);$$ Cauchy densities are an example. Suppose that $f_0 = g_q \circ \cdots \circ g_0$ lies in the compositional Hölder class $\mathcal{G}(q, \mathbf{d}, \mathbf{t}, \boldsymbol{\beta}, K)$ of Schmidt-Hieber: $d_0 = d$, $d_{q+1} = 1$, each $g_i$ maps $[a_i, b_i]^{d_i}$ into $[a_{i+1}, b_{i+1}]^{d_{i+1}}$ with $|a_i|, |b_i| \le K$, and each coordinate of $g_i$ depends on at most $t_i \le d_i$ of its variables and has $\beta_i$-Hölder norm at most $K$ (derivatives up to order $\lceil \beta_i \rceil - 1$, the last of them Hölder of order $\beta_i - \lceil \beta_i \rceil + 1$). Set $$\beta_i^* = \beta_i \prod_{j=i+1}^{q} \min(\beta_j, 1), \qquad \gamma = 2(1+\delta)(1+\kappa) + 1, \qquad \phi_n = \max_{0 \le i \le q} \Bigl(\frac{(\log n)^{\gamma}}{n}\Bigr)^{\beta_i^*/(2\beta_i^* + t_i)}.$$ For fixed $B > \|f_0\|_\infty$ let $C_B(f)(x) = \max(-B, \min(B, f(x)))$.

Castillo and Egels proved that for every $\alpha \in (0,1)$ the tempered posterior, in which the Gaussian likelihood is raised to the power $\alpha$, satisfies the conclusion below for $M$ large enough, and that the ordinary posterior does too when the prior is augmented by a suitable Gamma prior on the noise variance. They note that for heavy-tailed series priors in the Gaussian white noise model, the corresponding result for the ordinary posterior was shown by Agapiou and Castillo.

Problem. For every such model and prior, is there a constant $M > 0$ independent of $n$ such that $$\mathbb{E}_{f_0} \Pi_n\bigl(\|C_B(f) - f_0\|_{L^2(P_X)} > M \phi_n \bigm| X, Y\bigr) \longrightarrow 0 \qquad (n \to \infty),$$ where $\Pi_n(\cdot \mid X, Y)$ is the ordinary posterior, $$d\Pi_n(f \mid X, Y) \propto \exp\Bigl[-\frac{1}{2} \sum_{i=1}^{n} \bigl(Y_i - f(X_i)\bigr)^2\Bigr]\, d\Pi_n(f)?$$ The clipping $C_B$ is applied to posterior draws; the likelihood uses the unclipped network output.