Mod $p$ Iwasawa algebras of pro-$p$ Iwahori subgroups

Suppose $F$ is a finite unramified extension of ${\mathbb{Q}}_p$, and $G$ is the group of $F$-points of a split, connected, reductive group over $F$. Under a natural restriction on $p$, we determine the structure of the graded mod $p$ Iwasawa algebra ${\text{gr}}_{\mathfrak{m}}({\mathbb{F}}_p[\text{\hspace{0.17em}⁣}[I]\text{\hspace{0.17em}⁣}])$, where $I$ is a pro-$p$ Iwahori subgroup of $G$. We also determine its maximal commutative quotient, and relate these results to Gelfand–Kirillov dimensions of smooth mod $p$ representations of $G$.