Consecutive Piatetski-Shapiro primes conjecture

Let $k_0$ be a positive integer and let $c>1$ be non-integral. The Piatetski-Shapiro sequence is given by the integer parts $[n^c]$ for positive integers $n$. Consecutive Piatetski-Shapiro primes conjecture. There exist infinitely many $n$ such that

$$[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]$$

are all primes. This conjecture predicts arbitrarily long blocks of consecutive values in a Piatetski-Shapiro sequence that are simultaneously prime; the paper's result is motivated by this statement, while the conjecture itself remains open.