We construct a flexible (non-embedded) suspension with a hexagonal equator in Euclidean 3-space. It is known that the volume bounded by such a suspension is well defined and constant during the flex. We study its properties related to the Strong Bellows Conjecture which reads as follows: if a, possibly singular, polyhedron $\mathcal P$ in Euclidean 3-space is obtained from another, possibly singular, polyhedron $\mathcal Q$ by a continuous flex, then $\mathcal P$ and $\mathcal Q$ have the same Dehn invariants. It is well known that if $\mathcal P$ and $\mathcal Q$ are embedded, with the same volume and the same Dehn invariant, then they are scissors congruent.