  We study the Gram matrix determinants for the groups $S_n,O_n,B_n,H_n$, for
their free versions $S_n^+,O_n^+,B_n^+,H_n^+$, and for the half-liberated
versions $O_n^*,H_n^*$. We first collect all the known computations of such
determinants, along with complete and simplified proofs, and with
generalizations where needed. We conjecture that all these determinants
decompose as $D=\prod_\pi\phi(\pi)$, with product over all associated
partitions.
