Monotonicity conjecture for induced saturation in the hypergrid

Let $P$ be a poset, and consider the induced saturation function $\mathrm{sat}^{\star}([t]^n,P)$ whenever it is defined. Monotonicity conjecture. The function $(n,t)\mapsto\mathrm{sat}^{\star}([t]^n,P)$ is monotone non-decreasing in both $n$ and $t$ over its domain. The supplied text gives no resolution of this natural monotonicity claim.