Trace extension conjecture for forbidden posets

Let $P$ be a poset, let $e(P)$ be the largest integer $k$ such that, for every $j$ and $n$, the family

$$\bigcup_{i=1}^k\binom{[n]}{j+i}$$

is $P$-free, and let $Tr_{n-k}(n,P)$ denote the maximum size of a family whose traces on every $(n-k)$-subset are $P$-free. Trace extension conjecture. For every integer $k<e(P)$,

$$Tr_{n-k}(n,P)=(e(P)-k+o(1))\binom{n}{\lfloor n/2\rfloor}.$$

Moreover, if $k\ge e(P)$, then

$$Tr_{n-k}(n,P)=o\left(\binom{n}{\lfloor n/2\rfloor}\right).$$

This generalizes the preceding conjecture from ordinary forbidden subposet problems to traces. The displayed lower bound comes from consecutive levels, while the matching asymptotics and the vanishing assertion remain conjectural.