Wolstenholme's higher-power congruences for Gaussian integers

Let $i^2=-1$. For an integer $k\geq 1$ and a prime $p>17$, define

$$S_p^{(k)}=\sum_{\substack{1\leq n,m\leq p-1\\(p,n^2+m^2)=1}}\frac{1}{(mi+n)^k}.$$

Define $m(k)$ by

$$m(k)=\begin{cases}4 & \text{if } k\equiv 1\pmod 4,\\3 & \text{if } k\equiv 2\pmod 4,\\2 & \text{if } k\equiv 3\pmod 4,\\1 & \text{if } k\equiv 0\pmod 4.\end{cases}$$

Wolstenholme's higher-power congruences. For every such $k$ and $p$,

$$S_p^{(k)}\equiv 0\pmod {p^{m(k)}}.$$

Equivalently, the source writes $m(k)=4-(k-1)\pmod 4$. Computations establish the corresponding congruences for $k$ through $12$, but a general proof is not known; the difficulty is obtaining a general formula for the coefficient of the lowest power of $p$ in the relevant numerator.