Lifted toroidal-cycle conjecture for cycles trivial in the quotient

Let $K$ be a periodic cellular complex and let $G$ be a quotient space obtained from its translational periodicity. A toroidal cycle is a cycle in $G$ arising from the periodic directions, and a lift of such a cycle is a corresponding chain or cycle in $K$.

Lifted toroidal-cycle conjecture. All cycles in $K$ which are trivial in $G$ can be written in terms of lifts of toroidal cycles.

The claim would extend the known degree-one recovery of disappearing cycles and is illustrated by the periodic union of coordinate planes in $\mathbb{R}^3$, where the boundary of a unit cube vanishes in the quotient and is expressed using lifted toroidal $2$-cycles. Its validity in higher dimensions remains open.