Lu–Milans conjecture on Lubell functions of forbidden subposets

Let $P$ be a poset, and let $\bar{h}_n(\mathcal{F})$ denote the Lubell function of a family $\mathcal{F}$ of subsets of $[n]$. Define $\lambda_n(P)$ to be the maximum value of the Lubell function over families of subsets of $[n]$ that do not contain $P$ as a subposet. Lu–Milans conjecture. For every poset $P$,

$$\limsup_{n\to\infty} \lambda_n(P)<\infty.$$

Lu and Milans proposed this as a general finiteness conjecture for forbidden-subposet problems. The paper states that Méroueh verified it by proving an explicit Lubell-function bound for families avoiding $B_m$ and posets of bounded 2-dimension, so this conjecture is solved.