Gerbner–Nagy–Patkós–Vizer counting conjecture for tree posets

Let $P$ be a poset, and call a family $\mathcal F\subseteq 2^{[n]}$ $P$-free if it contains no copy of $P$. Let $La(n,P)$ denote the maximum size of a $P$-free family in $2^{[n]}$. Gerbner–Nagy–Patkós–Vizer conjecture. The number of $P$-free families in $2^{[n]}$ is

$$2^{(1+o(1))La(n,P)}.$$

This conjecture concerns the asymptotic enumeration of forbidden-poset-free families and is attributed in the source to Gerbner, Nagy, Patkós and Vizer; the supplied text gives no resolution status.