Weak compatibility conjecture for weighted Hardy-field averages

Let $\mathcal{H}$ be a Hardy field, let $f_1,\ldots,f_k,W\in\mathcal{H}$, and define

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

Property (WP) is that, for every $f\in\operatorname{span}^*(f_1,\ldots,f_k)$ and $p\in\mathbb{R}[t]$, either $|f(t)-p(t)|\ll1$ or $|f(t)-p(t)|\succ\log(W(t))$. Weak compatibility conjecture. Theorem main holds if its Property P is replaced by Property (WP). This would extend the paper's weighted convergence and recurrence theorem under a weaker separation condition; the source presents the implication as a conjecture and gives no resolution.