Odd-multiple valuation conjecture for inert Gaussian power sums

Let $p\ge7$ be a prime with $p\equiv3\pmod4$, and write $n=r(p-1)$ with $r\ge1$. Odd-multiple valuation conjecture.

$$v_p\!\bigl(\mathbf G_{r(p-1)}(p)\bigr)=\begin{cases}0,&r\text{ even},\\3,&r\text{ odd}.\end{cases}$$

Equivalently, among multiples of $p-1$,

$$v_p(\mathbf G_n(p))=0\quad\Longleftrightarrow\quad\operatorname{lcm}(4,p-1)\mid n,$$

for $p\equiv3\pmod4$, and outside this locus the valuation is $3$. The mod-$p$ behavior is known in the stated range, but the asserted valuation $3$ for odd multiples, including all $r\ge p$, remains conjectural.