The supersaturation conjecture for forbidden and induced subposets

Let $P$ be a poset and $\varepsilon>0$. Let $La(n,P)$ and $La^*(n,P)$ be the largest sizes of, respectively, $P$-free and induced $P$-free families in $2^{[n]}$. Let $e(P)$ and $e^*(P)$ be the corresponding consecutive-level parameters. Let $M(n,P)$ and $M^*(n,P)$ denote the maximum numbers of copies and induced copies, respectively, of $P$ in a family contained in $2^{[n]}$. The supersaturation conjecture. There exists $\delta>0$ such that every $\mathcal F\subseteq 2^{[n]}$ satisfying

$$|\mathcal F|\geq (e(P)+\varepsilon)\binom{n}{\lfloor n/2\rfloor}$$

contains at least $\delta M(n,P)$ copies of $P$; likewise, every family satisfying

$$|\mathcal F|\geq (e^*(P)+\varepsilon)\binom{n}{\lfloor n/2\rfloor}$$

contains at least $\delta M^*(n,P)$ induced copies of $P$.

This strengthens the asymptotic extremal conjecture by quantifying the number of forbidden configurations above the extremal threshold. The source presents it as a general conjecture and does not provide evidence of resolution.