Linear-growth dichotomy conjecture for induced saturation in the hypergrid

Let $P$ be a poset and fix $t\in\mathbb{Z}_{\geq 2}$. Write $\mathrm{sat}^{\star}([t]^n,P)$ for the induced saturation function of $P$ in the $t$-ary hypergrid. Linear-growth dichotomy conjecture. The function is either

$$\mathrm{sat}^{\star}([t]^n,P)=O(1)$$

or has the form

$$\mathrm{sat}^{\star}([t]^n,P)=cn+o(n)$$

for some constant $c=c(t,P)\in\mathbb{R}_{>0}$. This conjecture proposes a sharp bounded-versus-linear classification beyond the Boolean lattice; the supplied text gives no resolution.