Kalinin's polynomial congruence conjecture for Gaussian integers

Let $p>3$ be a prime, and define

$$g_p(x):=\prod_{1\leqslant m,n\leqslant p-1\atop p\nmid m^2+n^2}(x-(m+ni)).$$

Kalinin's conjecture. If $p\equiv1\pmod 4$, then

$$g_p(x)\equiv 1+\sum_{k=1}^{p-3}c_kx^{k(p-1)}\pmod p$$

for some $c_1,\ldots,c_{p-3}\in\mathbb Z$ with $c_{p-3}=1$. If $p\equiv3\pmod 4$, then

$$g_p(x)\equiv 1+x^{2(p-1)}+x^{4(p-1)}+\dots+x^{(p-1)^2}=\frac{1-x^{p^2-1}}{1-x^{2(p-1)}}\pmod p.$$

The conjecture extends Wolstenholme-type congruences from the integers to Gaussian integers. The paper presents it as a conjecture of N. Kalinin; the supplied text does not establish whether it has since been resolved.