Weak Bertrand conjecture for Gaussian lines

Let $L$ be a primitive Gaussian line, and let $\alpha_n$ denote the indexed Gaussian integers on $L$. For $\beta=x+iy\in\mathbb{Z}[i]$, write $N(\beta)$ for its Gaussian norm. Weak Bertrand conjecture for Gaussian lines. If $n>1$, then there is always at least one Gaussian prime on $L$ lying between $\alpha_n$ and $\alpha_{n+N(\alpha_n)}$. This is a norm-based analogue of Bertrand's postulate; the source explains that the norm interval may be more efficient than the $\nu$-interval for searching for Gaussian primes, but gives no resolution.