  We show that hyperoctahedral Whittaker functions---diagonalizing an open
quantum Toda chain with one-sided boundary potentials of Morse type---satisfy a
dual system of difference equations in the spectral variable. This extends a
well-known bispectral duality between the nonperturbed open quantum Toda chain
and a strong-coupling limit of the rational Macdonald-Ruijsenaars difference
operators. It is manifest from the difference equations in question that the
hyperoctahedral Whittaker function is entire as a function of the spectral
variable.
