Flatness in the totally even two-dimensional rational case

Determine whether the unframed universal deformation ring \(R_{\bar\rho}\) is flat over \(W(k)\) when \(K=\mathbb Q\), \(n=2\), the residual representation is continuous and absolutely irreducible over a finite field of characteristic \(2\), and complex conjugation acts trivially, \(\bar\rho(c)=I_2\).

Background

The proposed modified dimension conjecture predicts flatness when $\bar\rho(c)\ne I_2$, but the paper explicitly leaves open whether flatness should also be expected in the totally even case $\bar\rho(c)=I_2$.

The paper gives an example over $\mathbb Q$ with trivial residual complex conjugation for which the universal deformation ring is not a complete intersection. That example does not resolve the separate question of flatness.