Hasse-regularity conjecture for flag strata of Hodge-type Shimura varieties

Let $(G,\mu)$ be a cocharacter datum over $\mathbb{F}_q$, and let $(X,\zeta)$ satisfy the paper's stated assumptions. Let $Y$ be the flag space of $X$, stratified by flag strata $Y_w$, and let $z=\sigma(w_{0,I})w_0$, where $w_{0,I}$ and $w_0$ are the relevant longest Weyl-group elements. A flag stratum is Hasse-regular when the saturated cone generated by sections induced from the Hasse stack equals the saturated cone of all sections on its closure. Hasse-regularity conjecture. The flag stratum $Y_z$ is Hasse-regular. This property would control the weights of sections on the maximal open flag stratum through Hasse invariants. It is known in the Hilbert--Blumenthal example described in the paper, but the general assertion remains open.