The bottomless-rectangle linear decomposition conjecture

Let $\mathcal F$ be the family of bottomless rectangles in the plane, and let $m_k^*(\mathcal F)$ denote the corresponding threshold for decomposing coverings into $k$ coverings. Bottomless-rectangle conjecture.

$$m_k^*(\mathcal F)=O(k).$$

This is described as an especially difficult special case of the linear-growth conjecture and has received substantial attention; the source gives no resolution.