  For differential operators of Calabi-Yau type, Candelas, de la Ossa and van
Straten conjecture the appearance of $p$-adic zeta values in the matrix entries
of their $p$-adic Frobenius structure expressed in the standard basis of
solutions near a MUM-point. We prove that this phenomenon holds for simplicial
and hyperoctahedral families of Calabi-Yau hypersurfaces in $n$ dimensions, in
which case the Frobenius matrix entries are rational linear combinations of
products of $\zeta_p(k)$ with $1 < k < n$.
