Hardy-field prime recurrence conjecture for several functions

Let $k\in\mathbb{N}$ and let $a_1,\ldots,a_k\in\mathcal{H}$ be functions of polynomial growth. Assume that every non-trivial linear combination $a$ of $a_1,\ldots,a_k$ satisfies

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

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

$$\left\{n\in\mathbb{N}:\mu\left(A\cap T^{-\lfloor a_1(n)\rfloor}A\cap\dots\cap T^{-\lfloor a_k(n)\rfloor}A\right)>0\right\}$$

has non-empty intersection with the primes $\mathbb{P}$. The source presents this as a further conjecture motivated by known recurrence results along $\mathbb{N}$; it remains open in the source.