Frantzikinakis's logarithmic divergence conjecture for Hardy-field averages

Let $f_1,\ldots,f_k$ be functions from a Hardy field, and define

$$\operatorname{span}^*(f_1,\ldots,f_k)=\{c_1f_1(t)+\cdots+c_kf_k(t):(c_1,\ldots,c_k)\in\mathbb{R}^k\backslash\{0\}\}.$$

Let $(X,\mathcal{B},\mu,T)$ be an ergodic measure-preserving system and let $h_1,\ldots,h_k\in L^\infty(X)$. Frantzikinakis's conjecture. If $|f(t)-q(t)|/\log t\to\infty$ for every $q\in\mathbb{Z}[t]$ and $f\in\operatorname{span}^*(f_1,\ldots,f_k)$, then

$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T^{\lfloor f_1(n)\rfloor}h_1\cdots T^{\lfloor f_k(n)\rfloor}h_k=\prod_{i=1}^k\int_X h_i\,\mathrm{d}\mu$$

in $L^2(X)$. The conjecture predicts complete convergence to the product of means for sufficiently non-polynomial Hardy-field iterates; the source gives no resolution.