Hardy-sequence Szemerédi conjecture for polynomial-growth sequences

Let $\mathcal{H}$ denote the Hardy field under consideration, let $[x]$ be the integer part of $x$, and let $\bar d(\Lambda)$ denote the upper density of $\Lambda\subset\mathbb{Z}$. Let $a\in\mathcal{H}$ have polynomial growth and suppose that

$$|a(x)-cp(x)|\to\infty$$

for every $p\in\mathbb{Z}[x]$ and $c\in\mathbb{R}$. An arithmetic progression of length $\ell+1$ is a set of the form $\{m,m+r,\ldots,m+\ell r\}$. Hardy-sequence Szemerédi conjecture. For every $\ell\in\mathbb{N}$, every $\Lambda\subset\mathbb{Z}$ with $\bar d(\Lambda)>0$ contains an arithmetic progression of the form

$$\{m,m+[a(n)],m+2[a(n)],\ldots,m+\ell[a(n)]\}$$

for some $m\in\mathbb{Z}$ and $n\in\mathbb{N}$ with $[a(n)]\ne 0$. This would extend the known one-parameter result to arbitrary progression length, and is presented as a likely relaxation of the growth assumptions available when $\ell=1$; its status is not resolved in the source.