Weakly periodic tiling conjecture for exact clusters

Let $F\subseteq\mathbb Z^d$ be an exact cluster. An $F$-tiling $T$ is weakly periodic if there is a finite partition

$$T=T_1\sqcup\cdots\sqcup T_k$$

such that each $T_i$ is $1$-periodic.

Weakly periodic tiling conjecture. There exists a weakly periodic $F$-tiling of $\mathbb Z^d$; more precisely, there exists an $F$-tiling $T\subseteq\mathbb Z^d$ admitting such a finite partition.

This conjecture is proposed as a weaker replacement for the periodic tiling conjecture, which was announced to fail in sufficiently high dimensions. The analogous periodic tiling problem for $\mathbb Z^2\times(\mathbb Z/N\mathbb Z)$ remains unresolved, indicating that the proposed upgrading from weak to full periodicity is genuinely difficult.