On the dimension conjecture for global Galois deformation rings in characteristic two

Let $K$ be a number field, and let $S$ be a finite set of primes of $K$ containing all primes above $2$ and all archimedean primes. Let $G_{K,S}$ be the Galois group of the maximal algebraic extension of $K$ unramified outside $S$. We prove two refined lower bounds for the Krull dimension of unrestricted global Galois deformation rings of continuous absolutely irreducible representations of $G_{K,S}$ over finite fields of characteristic $2$. As applications, we construct the first counterexamples to the dimension conjecture in every dimension $n\ge 2$ over every number field with a real prime. Over every totally real number field, these constructions also yield the first examples of such universal deformation rings that fail to be complete intersections. Furthermore, we propose a modified dimension conjecture for $K=\mathbb{Q}$ and $n=2$.