Lexicographic initial-segment conjecture for Frankl families

For a set $S\in\binom{[n]}{k}$, let $[S]$ denote the initial segment through $S$ in the lexicographic order on $\binom{[n]}{k}$. Let $FC(k,n)$ be the least integer such that every union-closed family containing $FC(k,n)$ distinct $k$-sets of an $n$-set satisfies Frankl's conjecture with an element of that $n$-set. An FC-family is a family of sets that satisfies Frankl's conjecture, and the universe size of a family is the cardinality of the union of its members.

Lexicographic initial-segment conjecture. For fixed $n>k\geq3$, if $[S_m]$ is an FC-family for some positive integer $m$ and has universe size $n$, then

$$FC(k,n)\leq m.$$

The claim predicts that a lexicographic initial segment that is an FC-family supplies an upper bound for the extremal quantity $FC(k,n)$. It is presented as a conjecture based on experimental results; no resolution is given in the supplied text.