  According to work of Hartley and Kawauchi in 1979 and 1980, the Conway
Polynomial of all negative amphicheiral knots and strongly positive
amphicheiral knots factors as $\phi(z)\phi(-z)$ for some $\phi(z)\in\mathbb
Z[z]$. Moreover, a 2012 example due to Ermotti, Hongler and Weber shows that
this is not true for general amphicheiral knots. On the other hand, in 2006 the
first author made a conjecture equivalent to saying that the Conway polynomial
of all amphicheiral knots splits as $\phi(z)\phi(-z)$ in the ring $\mathbb
Z_4[z]$. In this paper, we establish this conjecture for all periodically
amphicheiral knots built from braids, where the period preserves the braid
structure. We also give counterexamples to conjectures on the leading
coefficient of the Conway polynomial of an amphicheiral knot due to Stoimenow.
