Flatness conjecture for spin local models of split even orthogonal groups

Let $F$ be a discretely valued field with ring of integers $\CO$, and let $G$ be the split even orthogonal group of rank $2n$. Let $\CL$ be a self-dual lattice chain, and let $\RM^\pm_\CL$ be its spin local model, defined as the closed subscheme of the naive local model whose exterior-power condition uses one of the two eigenspaces $W_\pm$ of the involution on $\bigwedge_F^n V$. Let $\RM^{\pm\loc}_\CL$ be the schematic closure of the generic fiber $\RM^\pm_\CL\otimes F$ in $\RM^\pm_\CL$. Flatness conjecture. The spin local model $\RM^\pm_\CL$ is $\CO$-flat. Equivalently,

$$\RM^\pm_\CL=\RM^{\pm\loc}_\CL.$$

This conjecture asks whether the spin condition gives the flat closure of the generic orthogonal Grassmannian in every self-dual lattice-chain case. The paper's introduction reduces the claim to reducedness of the special fiber, but the supplied text gives no resolution.