  We collect evidence in support of a conjecture of Griffiths, Green and Kerr
on the arithmetic of extension classes of limiting mixed Hodge structures
arising from semistable degenerations over a number field. After briefly
summarizing how a result of Iritani implies this conjecture for a collection of
hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the
authors investigate a sequence of (non-hypergeometric) examples in dimensions 1
through 6 arising from Katz's theory of the middle convolution. A crucial role
is played by the Mumford-Tate group (of type G2) of the family of 6-folds, and
the theory of boundary components of Mumford-Tate domains.
