The new-condition flatness conjecture for ramified unitary local models

Let $F/F_0$ be a ramified quadratic extension of discretely valued non-Archimedean fields with common residue field of characteristic not $2$, let $n\geq 2$, let $(r,s)$ be any signature with $r+s=n$, and let $I\subset\{0,\dotsc,\lfloor n/2\rfloor\}$ be nonempty and satisfy

$$n\text{ is even and }\lfloor n/2\rfloor-1\in I\implies \lfloor n/2\rfloor\in I.$$

Let $M_I$ be the refinement of the naive local model defined by the new condition, and let $E$ be the reflex field. The new-condition flatness conjecture. For any such signature and $I$,

$$M_I\text{ is flat over }\operatorname{Spec}{\mathscr{O}}_E,$$

or equivalently,

$$M_I=M_I^{\mathrm{loc}}.$$

The new condition refines the spin and Kottwitz conditions and is designed to characterize the honest local model; the source presents this flatness and equality as the central conjecture of the paper.