Shimoda’s conjecture on complete-intersection primes implying dimension at most two

Prove the conjecture that any Noetherian local ring in which every prime ideal other than the maximal ideal is a complete intersection must have Krull dimension at most two.

Background

Alongside Sally’s problem, the authors reference a related conjecture posed by Shimoda concerning the implications of all non-maximal prime ideals being complete intersections.

This conjecture is presented as an apparently simpler statement whose resolution would clarify how stringent complete-intersection behavior of prime ideals constrains the dimension of a Noetherian local ring.

References

Or to undertake the apparently more simple conjecture stated by Shimoda on whether a noetherian local ring such that all its prime ideals different from the maximal ideal are complete intersection, has dimension at most two (see, e.g., [9]).

— Explicit minimal generating sets of a family of prime ideals with unbounded minimal number of generators in a three-dimensional power series ring  (2604.00638 - González et al., 1 Apr 2026) in Section 1. Introduction