The monotonicity conjecture for grid forbidden-subposet densities

Let $P$ be a poset of dimension at most $d$, and let $\pi_{d,P}$ and $\pi^*_{d,P}$ denote the weak and strong grid density parameters. The grid-density monotonicity conjecture.

$$\pi_{d,P}\geq\pi_{d+1,P}\qquad\text{and}\qquad\pi^*_{d,P}\geq\pi^*_{d+1,P}.$$

This asserts that both density parameters are monotone decreasing as the grid dimension increases. It is posed as an open problem in the paper.