The odd-modulus group-to-cyclic solution conjecture for linear systems

Let $d>1$ be an odd integer. A linear system over $\mathbb{Z}_d$ is a system of linear constraints with coefficients and right-hand sides in $\mathbb{Z}_d$; a solution in a group $G$ means an assignment satisfying the constraints in the paper's group-theoretic sense. Odd-modulus solution conjecture. Any linear system over $\mathbb{Z}_d$ admitting a solution in $G$ also admits a solution in $\mathbb{Z}_d$. This conjecture predicts that, for odd moduli, group-valued solutions do not provide more solvability than cyclic solutions; the paper reports significant evidence but does not establish the claim.