Flatness conjecture for the local-model weight morphisms

Let $X_L$ be the local model with closed subschemes $X_w$, let $\widehat{\mathcal X}^{\flat}_{L,\widehat y}$ and $\widehat{\mathcal X}^{\flat}_{L,w,\widehat y}$ be the corresponding formal schemes, and let $\widehat{\mathfrak t}_L$ be the formal Cartan scheme. For $i=1,2$, write $\kappa_i$ and $\kappa_{w,i}$ for the induced morphisms to $\widehat{\mathfrak t}_L$. Flatness conjecture. The morphisms

$$\kappa_i:\widehat{\mathcal X}^{\flat}_{L,\widehat y}\rightarrow\widehat{\mathfrak t}_L,$$

respectively

$$\kappa_{w,i}:\widehat{\mathcal X}^{\flat}_{L,w,\widehat y}\rightarrow\widehat{\mathfrak t}_L,$$

are flat. The paper proves the analogous assertion when $r=1$, but the general flatness statement is presented as a conjecture.