Uniform generation conjecture for minors of finite abelian p-groups

Let $\mathbf{A}$ be a finite abelian $p$-group, and let $\mathbf{R}$ be a ring, considered as a regular $\mathbf{R}$-module, such that $p$ is invertible in $\mathbf{R}$. Let $\mathcal O_{\mathbf{A},\mathbf{R}}$ denote the set of operations from $\mathbf{A}$ to $\mathbf{R}$, and let an $(\mathbf{A},\mathbf{R})$-minor be an operation obtained by the corresponding minor construction.

Uniform generation conjecture. There is a $k\in\mathbb{N}$ such that $\mathcal O_{\mathbf{A},\mathbf{R}}$ is uniformly generated by $k$-ary $(\mathbf{A},\mathbf{R})$-minors.

The conjecture proposes the uniform-generation property needed to extend the paper's method beyond the cases already proved, including the finite-vector-space to coprime-module setting. Its general validity is left open.