  In 2007, Y. Shimoda, in connection with a long-standing question of J. Sally,
asked whether a Noetherian local ring, such that all its prime ideals different
from the maximal ideal are complete intersections, has Krull dimension at most
two. In this paper, having reduced the conjecture to the case of dimension
three, if the ring is regular and local of dimension three, we explicitly
describe a family of prime ideals of height two minimally generated by three
elements. Weakening the hypothesis of regularity, we find that, to achieve the
same end, we need to add extra hypotheses, such as completeness, infiniteness
of the residue field and the multiplicity of the ring being at most three. In
the second part of the paper we turn our attention to the category of standard
graded algebras. A geometrical approach via a double use of a Bertini Theorem,
together with a result of A. Simis, B. Ulrich and W.V. Vasconcelos, allows us
to obtain a definitive answer in this setting. Finally, by adapting work of M.
Miller on prime Bourbaki ideals in local rings, we detail some more technical
results concerning the existence in standard graded algebras of homogeneous
prime ideals with an "excessive" number of generators.
