Conjecture on Legendre-symbol control of Gaussian-prime walk equivalence

Let $a+bi$ be a Gaussian prime and let $\frac{q}{p}$ be the Legendre symbol appearing in the comparison of the two Gaussian-prime ratios $\frac{a+bi}{\beta+\beta i}$ and $\frac{a+bi}{\beta+\beta i}$; the source's displayed expressions are retained below in the claim. Equivalence conjecture. The equivalence between the two corresponding bracketed quantities depends only on the value of the Legendre symbol $\frac{q}{p}$. In particular,

$$\left[\dfrac{a+bi}{\alpha+\beta i}\right] = \left[\dfrac{a+bi}{\beta+\alpha i}\right]$$

if $\frac{q}{p}=1$, and

$$\left[\dfrac{a+bi}{\alpha+\beta i}\right] \not= \left[\dfrac{a+bi}{\beta+\alpha i}\right]$$

if $\frac{q}{p}\not=1$. This conjecture is proposed to explain the observed strong positive and negative correlations in the Gaussian-prime random walks; the source does not establish its status beyond leaving the equivalence conditions as a conjecture.