The finite Turán threshold conjecture for posets

Let $P$ be a finite poset. For each $n$, let ${\rm La}^{*}(n,P)$ be the maximum size of a family $\mathcal{F}\subseteq 2^{[n]}$ containing no induced copy of $P$, and define the induced Turán threshold by

$$\pi^{*}(P)=\limsup_{n\to\infty}\frac{{\rm La}^{*}(n,P)}{\binom{n}{\lfloor n/2\rfloor}}.$$

Finite Turán threshold conjecture. Every poset has a finite Turán threshold, that is, $\pi^{*}(P)<\infty$ for every finite poset $P$.

The paper proves finiteness for series-parallel posets and posets of height $2$. The conjecture asks for the corresponding result for arbitrary finite posets.