Primes in irrational Beatty sequences

Let $\alpha>0$ be an irrational number and let $\beta$ be a real number. The expression $\floor{\alpha p+\beta}$ denotes the largest integer not exceeding $\alpha p+\beta$. Primes in irrational Beatty sequences conjecture. There exist infinitely many primes $p$ such that $\floor{\alpha p+\beta}$ is also prime. This is motivated by the binary Goldbach and twin primes conjectures and concerns simultaneous primality along an irrational Beatty sequence. The supplied source does not indicate whether the conjecture has been resolved.