Morris's conjecture on the threshold for 3-uniform FC-families

For integers $n\geqslant 4$, let $FC(3,n)$ denote the threshold quantity for 3-uniform families on an $n$-element ground set as defined in the paper. Morris's conjecture.

$$FC(3,n)=\left\lfloor \frac{n}{2}\right\rfloor+1.$$

The conjecture was suggested by the paper's discussion of known $FC$-families and is presented as an open direction following partial results.