Existence of non-flat unrestricted global deformation rings in characteristic two

Determine whether there exists a continuous absolutely irreducible representation \(\bar\rho:G_{K,S}\to\mathrm{GL}_n(k)\), with \(k\) of characteristic \(2\), \(n\ge 2\), and \(S\) containing all primes above \(2\) and all archimedean primes, whose unrestricted universal deformation ring \(R_{\bar\rho}\) is not flat over \(W(k)\), equivalently, has nonzero \(2\)-torsion.

Background

The paper constructs characteristic-two counterexamples to the original dimension conjecture and examples of universal deformation rings that are not complete intersections, not Cohen–Macaulay, and not equidimensional. These results do not determine whether unrestricted universal deformation rings can fail to be flat over the Witt vectors.

The question is posed under the standard hypotheses that the residual representation is continuous and absolutely irreducible, has dimension $n\ge2$, and that the ramification set contains all primes above $2$ and all archimedean primes. A positive answer would be equivalent to the existence of nonzero $2$-torsion in such a deformation ring.