Hardy-field prime recurrence conjecture for one function

Let $\mathcal{H}$ be the Hardy field under consideration, let $a\in\mathcal{H}$ have polynomial growth, and assume

$$\lim_{t\to+\infty}|a(t)-cp(t)|=+\infty$$

for every $c\in\mathbb{R}$ and $p(t)\in\mathbb{Z}[t]$. Let $(X,\mathcal{X},\mu,T)$ be a measure-preserving system, let $A\subset X$ have positive measure, and let $k\in\mathbb{N}$. Hardy-field prime recurrence conjecture. The set

$$\left\{n\in\mathbb{N}:\mu\left(A\cap T^{-\lfloor a(n)\rfloor}A\cap\dots\cap T^{-k\lfloor a(n)\rfloor}A\right)>0\right\}$$

has non-empty intersection with the primes $\mathbb{P}$. The source says the result is close to being established but notes examples not covered by the paper's methods; the conjecture is therefore open.