  The Ising spin-glasses are investigated on three dual pairs of hierarchical
lattices, using exact renormalization-group transformation of the quenched bond
probability distribution. The goal is to investigate a recent conjecture which
relates, on such pairs of dual lattices, the locations of the multicritical
points, which occur on the Nishimori symmetry line. Towards this end we
precisely determine the global phase diagrams for these six hierarchical
spin-glasses, using up to 2.5 x 10^9 probability bins to represent the quenched
distribution subjected to an exact renormalization-group transformation. We
find in all three cases that the conjecture is realized to a very good
approximation, even when the mutually dual models belong to different spatial
dimensionalities d and have different phase diagram topologies at the
multicritical points of the conjecture and even though the contributions to the
conjecture from each lattice of the dual pair are strongly asymmetric. In all
six phase diagrams, we find reentrance near the multicritical point. In the
models with d=2 or 1.5, the spin-glass phase does not occur and the phase
boundary between the ferromagnetic and paramagnetic phases is second order with
a strong violation of universality. PACS numbers: 75.10.Nr, 64.60.Kw, 05.45.Df,
05.10.Cc
