Strong Bertrand conjecture for Gaussian lines

Let $L$ be a primitive Gaussian line, and let $\alpha_n$ denote the indexed Gaussian integers on $L$. For $\beta=x+iy\in\mathbb{Z}[i]$, define

$$\nu(\beta)=\frac{N(\beta)}{\operatorname{gcd}(x,y)}.$$

Here $N$ is the Gaussian norm. Strong Bertrand conjecture for Gaussian lines. If $n>1$, then there is always at least one Gaussian prime on $L$ lying between $\alpha_n$ and $\alpha_{n+\nu(\alpha_n)}$. This conjecture generalizes Bertrand's postulate from the real line to primitive Gaussian lines; its status is not resolved in the source.