The counting conjecture for forbidden and induced subposets

Let $P$ be a finite poset. Let $e(P)$ and $e^*(P)$ be the consecutive-level parameters for, respectively, $P$-free and induced $P$-free families in $2^{[n]}$. The counting conjecture. The number of $P$-free families in $2^{[n]}$ is

$$2^{(e(P)+o(1))\binom{n}{\lfloor n/2\rfloor}},$$

and the number of induced $P$-free families is

$$2^{(e^*(P)+o(1))\binom{n}{\lfloor n/2\rfloor}}.$$

The lower bounds follow from taking subfamilies of consecutive middle levels. The source discusses known results for $P_2$ but gives no general resolution of these assertions.