  We consider all irreducible rank-4 hypergeometric local systems defined over
$\mathbb{Q}$ that support a rational one-dimensional variation of Hodge
structures of weight 3 and Hodge vector $(1,1,1,1)$. Up to a natural
equivalence there are only 47 cases. The first 14 cases have maximally
unipotent monodromy at one point and have been extensively studied in the
literature. We show that all 47 local systems are associated to families of
generically smooth threefolds and we analyze the geometry and arithmetic at
their conifold point.
