Frantzikinakis's prime progressions with fractional-power differences conjecture

Let $k\in\mathbb{N}$ and let $c$ be a positive real number. Write $\mathbb{P}$ for the set of primes. Frantzikinakis's conjecture. For infinitely many pairs $(p,m)\in\mathbb{P}\times\mathbb{N}$,

$$\{p,p+\lfloor m^c\rfloor,p+2\lfloor m^c\rfloor,\ldots,p+k\lfloor m^c\rfloor\}\subseteq\mathbb{P}.$$

Moreover, for any positive real numbers $c_1,\ldots,c_k$, there are infinitely many pairs $(p,m)\in\mathbb{P}\times\mathbb{N}$ such that

$$\{p,p+\lfloor m^{c_1}\rfloor,p+\lfloor m^{c_2}\rfloor,\ldots,p+\lfloor m^{c_k}\rfloor\}\subseteq\mathbb{P}.$$

The source presents this as a problem posed by Frantzikinakis, generalizing an earlier result in two directions: arithmetic progressions whose common difference is a fractional-power value, and configurations involving several fractional powers. It remains open.