Asymptotic growth conjecture for $(P,\phi)$-Tamari lattices

Let $L_n^d$ be the lattice of $d$-torsion classes associated with the $(P,\phi)$-Tamari lattice, and let $J(os_n^{d+1})$ denote the set of order ideals of the poset $os_n^{d+1}$. For fixed $n$, consider the limit as $d$ tends to infinity. Asymptotic growth conjecture. For $n$ fixed,

$$|L_n^d|=\mathcal{O}_{d\rightarrow \infty}\big(|J(os_n^{d+1})|\big).$$

This conjecture proposes that the size of $L_n^d$ is bounded asymptotically by the number of order ideals of $os_n^{d+1}$ as the dimension parameter grows, complementing the paper's exact enumeration results for small values of $n$.