Module-support conjecture for the q=1,2 rows of the LHS spectral sequence

Let $G$ be a 3D space group, and consider the standard Lyndon–Hochschild–Serre spectral sequence for its translation extension. At the $E_2$ page, examine the module structures in the rows with $q=1$ and $q=2$. Module-support conjecture. The module generator of each of the $q=1$ and $q=2$ rows can occur only in the columns $p=0$ or $p=1$. This proposed restriction is intended to control the possible degrees of cohomology-ring generators through the spectral-sequence calculation, but remains conjectural.