Hardy-sequence patterns in the primes conjecture

Let $\mathcal{H}$ denote the Hardy field under consideration, let $[x]$ be the integer part of $x$, and let “the growth assumptions of Theorem A (or Conjecture A)” refer to the hypotheses imposed there on $a$. Hardy-sequence patterns in the primes conjecture. If $a\in\mathcal{H}$ satisfies the growth assumptions of Theorem A (or Conjecture A), then the prime numbers contain arbitrarily long arithmetic progressions of the form

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

with the corresponding parameters as in Conjecture A. This is motivated by the role of Szemerédi-type results in establishing arithmetic and polynomial progressions in the primes, but the source gives no resolution.