Morris's minimal-generator extension conjecture for non-FC families

Let $\mathcal{B}$ be a union-closed family with minimal generating family $\mathcal{S}$. For some $B\in\mathcal{S}$ and an element $i$, let $\mathcal{B}^{\prime}$ be the family generated by

$$(\mathcal{S}\setminus\{B\})\cup\{B\cup\{i\}\}.$$

Minimal-generator extension conjecture. If $\mathcal{B}$ is not $FC$, then $\mathcal{B}^{\prime}$ is not $FC$ either. This conjecture proposes that replacing a minimal generator by its one-element extension preserves failure of the $FC$ property; the paper presents it as a final open conjecture motivated by the examples known there.