The mod-4 valuation law for Gaussian power sums

Let $p$ be an odd prime and define

$$\mathbf G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n\in\mathbb Z[i],$$

with $v_p(x+yi):=\min\{v_p(x),v_p(y)\}$. The mod-4 valuation conjecture. For $1\le n\le p-2$,

$$v_p(\mathbf G_1(p))=1,\qquad v_p(\mathbf G_2(p))=2,\qquad v_p(\mathbf G_3(p))=3,$$

and for $4\le n\le p-2$,

$$v_p(\mathbf G_n(p))=\begin{cases}1,&n\equiv0\pmod 4,\\2,&n\equiv1\pmod 4,\\3,&n\equiv2\pmod 4,\\4,&n\equiv3\pmod 4.\end{cases}$$

These formulas are suggested by the computed data and describe the observed valuation pattern for small exponents; their general validity is unresolved.