Consecutive-level minimizer conjecture for tree-shaped posets

Let $T$ be a poset whose Hasse diagram is a tree. Let $La(n,T)$ be the maximum size of a $T$-free family in $2^{[n]}$, and let $e(T)$ be the largest integer such that a family from $\Sigma^*(n,e(T))$ is $T$-free. For a positive integer $E$, consider families of size $La(n,T)+E$ and count their copies of $T$.

Tree-poset consecutive-level minimizer conjecture. There exists a constant $c=c(T)$ with $0<c\le1$ such that, if $E\le c\binom{n}{\lfloor n/2\rfloor}$, then some family $\mathcal F\subseteq2^{[n]}$ minimizing the number of copies of $T$ among families of size $La(n,T)+E$ contains sets of only $e(T)+1$ different sizes, and those sizes are consecutive integers.

The conjecture seeks a structural description of minimizers just above the extremal size. Even for tree posets, the exact value of $La(n,T)$ is not known in general, and the proposed minimizer structure is left open.