Lucas congruence for Gaussian binomial coefficients

For integers $A\geq C\geq 1$ and $B\geq D\geq 1$, define the Gaussian binomial coefficient by

$$\left[\substack{A+Bi\\ C+Di}\right]:=\frac{\prod\limits_{\substack{0\leq n\leq C-1,\\0\leq m\leq D-1}}(A+Bi-(n+mi))}{\prod\limits_{\substack{1\leq n\leq C,\\1\leq m\leq D}}(n+mi)}.$$

Lucas congruence for Gaussian binomial coefficients. For positive integers $A,B,C,D$ for which these coefficients are defined,

$$\left[\!\substack{pA+pBi\pC+pDi}\!\right]\equiv\left[\!\substack{A+Bi\C+Di}\!\right]\pmod {p^3}.$$

This is proposed as a possible analogue of Lucas' theorem. The source notes that these coefficients are not generally Gaussian integers and that little is known about their properties, so the congruence remains open.