On sums of primes from Beatty sequences

Let $k \ge 2$ and $\alpha_1, \beta_1, ..., \alpha_k, \beta_k$ be reals such that the $\alpha_i$'s are irrational and greater than 1. Suppose further that some ratio $\alpha_i/\alpha_j$ is irrational. We study the representations of an integer $n$ in the form $$p_1 + p_2 + ... + p_k = n,$$ where $p_i$ is a prime from the Beatty sequence $$\mathcal B_i = \left\{n \in \mathbb N : n = [ \alpha_i m + \beta_i ] \text{for some} m \in \mathbb Z \right\}.$$